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Quantum wave packet revivals in IR + X-ray pump–probe spectroscopy
Chemical Physics Letters 405 (2005) 398–403 www.elsevier.com/locate/cplett
Quantum wave packet revivals in IR + X-ray pump–probe spectroscopy F.F. Guimara˜es
˚ gren , F. Gelmukhanov a, A. Cesar b, H. A
a,b,*
a
a
b
Theoretical Chemistry, Roslagstullsbacken 15, Royal Institute of Technology, S-106 91 Stockholm, Sweden Departamento de Quı´mica, Universidade Federal de Minas Gerais, Av. Antonio Carlos 6627, CEP-31270-901, Belo Horizonte, Minas Gerais, Brazil Received 10 January 2005; in final form 14 February 2005 Available online 16 March 2005
Abstract The wave packet revivals constitute a central concept of X-ray spectroscopy with ultra-high spectral resolution. The revival phenomenon allows to resolve the anharmonical shift or rotational structure by means of time dependent measurements and makes X-ray pump–probe spectroscopy a powerful technique to study long-term dynamics of molecules in different phases. We study the revivals referring to the X-ray absorption spectrum of the NO molecule driven by strong infrared pulse. It is shown that the phase sensitive trajectories of the center of gravity of the wave packets and the X-ray spectra copy each other. 2005 Elsevier B.V. All rights reserved.
1. Introduction Advances in the physics and chemistry of laser interactions with atoms and molecules have brought the concept of wave-packets and their dynamics into the limelight. One of the major reasons for studying wave packet dynamics in the context of molecules is related to laser catalysis [1,2] and the control of chemical reactions [3–8] by careful application of pulses of light with optimal frequency, intensity, duration, and timing. Among these studies a special attention has been paid to the long-term dynamics related to quantum wave packet revivals. Classical as well as quantum dynamics of systems of coupled oscillators with slightly different frequencies experience fast oscillations with vibrational frequency and slow modulations. These modulations, defined by the differences between the frequencies of individual oscillators, are named revivals. Revival phenomena are *
Corresponding author. Fax: +46 8 5537 8590. E-mail addresses: [email protected], [email protected] (F.F. Guimara˜es). 0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.02.061
observed in many branches of physics. The first studies related to the revival time are traced to the Poincare recurrences for a rotation map [9], where the revival is analyzed in classical systems. Quantum revivals are usually associated with the dynamics of wave packets (WP) invented in physics by Schro¨dinger [10]. Quantum mechanical dephasing caused by anharmonicity (nonequidistant spectrum) leads to a delocalization of the initial wave packet which regains its initial shape after the revival period. The wave packet revival, predicted by Parker and Stroud [11], was experimentally confirmed a few years later [12]. Wave packet revivals are recognized to be important, for instance, in the motion of Rydberg wave packets [11,13], for rotational and vibrational degrees of freedom [14–17], in core-excited states [18,19], for pulse shaping [20,21], and for isotope separation [22]. The current state of theoretical and experimental studies of the wave packet revivals in atomic, molecular, condensed matter and optical systems has been reviewed by Robinett [23] a short time ago. The quantum revival is essentially a coherent phenomenon related to the dynamics of coherent superposition of quantum states. The wave packets can be
2. Physical picture of quantum revivals in an X-ray absorption spectrum We consider molecules that interact with IR pump field (L) and probe X-ray radiation (X) (see Fig. 1): Ea ðtÞ ¼ Ea ðtÞ cosðxa t ka R þ ua Þ;
a ¼ L; X :
ð1Þ
The pump and probe fields Ea(t) = eaEa(t) are characterized by the polarization ea, the wave vectors envelopes ka, the frequencies xa and the phases ua. Atomic units are assumed everywhere in this Letter. The IR radiation influences the X-ray absorption in two different ways: due to the population of vibrational levels (m) and through the IR induced coherence between these levels. The coherent IR light mixes different vibrational states and creates their coherent superposition or the wave packet. Since the wave packet is not an eigenfunction of the ground state Hamiltonian, it starts to move. The coherently created wave packet performs the back and forth propagation in the potential well of the ground electronic state. X-ray snapshots of the nuclear
534.0
Energy (eV)
generated in the potential energy surface due to optical [14] or X-ray transitions [24] as well as by means of vibrational–rotational transitions in the field of an infrared (IR) laser. The coherent properties of wave packets induced by an IR field are related directly to their phases and gives the opportunity to construct different superpositions of quantum states which are sensitive to the phase of the pump radiation. This constitutes the background of phase sensitive IR–X-ray pump–probe spectroscopy [25–28]. The main aim of our article is to study the role of quantum revivals in IR–X-ray pump–probe spectroscopy. Contrary to optical and IR spectroscopies, conventional X-ray spectroscopy has rather poor spectral resolution, mainly due to the rather large lifetime broadening and because of the quite large instrumental broadening. We show that the center of gravity of X-ray probe spectrum experiences modulation in the time domain with the revival period inversely proportional to the anharmonicity constant which is smaller than the lifetime broadening of X-ray resonance. Thus, the revival phenomenon makes this pump–probe setup very promising in X-ray spectroscopy with ultra-high resolution, far beyond the limitations due to the lifetime broadening. Measurements of the revival period give direct information about hyperfine structure of the molecular spectrum. We will also show how the X-ray pump–probe spectroscopy can be used in studies of inter- and intramolecular interactions. We deal with revivals related to the anharmonicity of the interatomic potential in the ground electronic state.
399
Ω (eV)
F.F. Guimara˜es et al. / Chemical Physics Letters 405 (2005) 398–403
533.0
A
532.0
B
531.0 Absorption probability
5.0 4.0 3.0 2.0 1.0 0.0 1.5
A
2.0
B
2.5
3.0
3.5
r (a.u.) Fig. 1. Scheme of OK X-ray absorption of NO driven by a strong IR field. At a certain instant a short X-ray pulse promotes the wave packet (A or B) to a certain point of the core excited potential. The X-ray absorption profiles A and B show the dependence of the X-ray spectra on the delay between X-ray and IR pulses.
wave packet at different site positions yield the X-ray spectra (see Fig. 1). This technique maps the trajectory of the WP and is useful in studies of intramolecular interaction in excited states [28,27]. The ground-states nuclear wave packet obeys the Schro¨dinger equation i
o ^ þ C /ðtÞ ¼ ½H 0 ðd EL ðtÞÞ cosðxL t þ uL Þ/ðtÞ; ot ð2Þ
^ and with the Hamiltonian H0, the relaxation matrix C initial condition j/(0)æ = j0æ. We neglect the spatial phases kL Æ R and kX Æ R the role of which was already discussed [25]. Before embarking on the details of the calculations, we employ simple arguments based on the rotating wave approximation (xL x10): when the IR pulse leaves the system the dynamics of the wave packet (2) is straightforwardly described by X jmiam eðim þCm Þðtt0 Þ ; j/ðtÞi ¼ m
am ¼ i e
iuL
Z
1
dt1 ðdm;m1 EL ðt1 ÞÞ 1 e½iðxm;m1 xL ÞþCm Cm1 ðt1 t0 Þ am1 ðt1 Þ
¼ jam jeiðmuL þvm Þ ; where vm is the intrinsic phase of the vibrational state m, which depends on the frequency and the shape of the IR pulse; t0 is the time when the IR pulse leaves the system, xm,m1 = m m1; m and jmæ are the vibrational energy and eigenvector of the ground electronic state, and; Cm is the decay rate of the vibrational level m (C0 = 0). It can be seen that even after that the IR pulse
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left the system the wave packet continues to be coherent and keeps the memory about the laser phase uL during the lifetime of the vibrational levels, C1 m . The physical understanding of the revival phenomenon can be achieved by considering the IR field of moderate intensity which mixes only the first three vibrational levels, m = 0,1,2. It is instructive to analyze the trajectory of the wave packet which is characterized by its center of gravity hrðtÞi ¼ h/ðtÞjrj/ðtÞi ¼ re þ 2Re q10 eðix10 þC10 Þðtt0 Þ r01 þq21 eðix21 þC21 Þðtt0 Þ r12 :
ð3Þ
Here, qmm1 ¼ am am1 is the density matrix, rmm1 ¼ hmjrjm1 i, re is the equilibrium interatomic separation, and Cmm1 ¼ Cm þ Cm1 . When the molecule is embedded in a bath (gas or liquid), it experiences collisions which quench the coherence qm,m1 with the rate cm,m1 c. Such a dephasing increases the decay rate of the coherence Cm;m1 ¼ Cm þ Cm1 þ c:
ð4Þ
To be specific, we assume that the density of the sample is quite high: Cm,m1 c. The resulting expression for the center of gravity (3) can be written as follows pffiffiffiffiffiffiffiffiffiffiffiffiffi hrðtÞi ¼ re þ 2r01 q00 q11 ð1 fÞ cosðx10 t0 þ u10 Þ t0 0 þ 2f cosðx10 t0 þ uþ Þ cos 2p þ u ect ; TR ð5Þ 0
where t = t t0 P 0, u± = (u10 ± u21)/2, u10 = uL + u1 u0, u21 = uL + u2 u1. We used here the harmonic approximation pffiffifor ffi the ratio of transition dipole moments, r12 =r01 ¼ 2, and introduced the auxiliary parameter sffiffiffiffiffiffiffiffiffi 2q22 : ð6Þ f¼ q00 The center of gravity of the wave packet (5) experiences fast oscillations with the period inversely proportional to the vibrational frequency x10 which are modulated by slow oscillations with the period equal to the revival time TR ¼
4p 2p ¼ x10 x21 x10 xe
ð7Þ
inversely proportional to the anharmonicity constant x10 xe = 13.71 cm1 = 0.0017 eV. This equation follows from the expression for the eigenvalues of the Morse oscillator: m = x10(m + 1/2)x10xe(m + 1/2)2. The important characteristic of the WP revival is its contrast, (Ær(t)æmax re)/jÆr(t)æmin rej. When the population of the vibrational level m = 2 is small the contrast is related directly to the parameter f (6)
hrðtÞimax re 1þf hrðtÞi r ¼ j1 fj ; e min
f 1:
ð8Þ
Our simulations show that for large f this ratio becomes sensitive to the phases u10 and u21. The wave packet revivals can be observed making use of X-ray absorption as is illustrated in Fig. 1. The potentials of the ground and core-excited states usually differ, which implies that the X-ray absorption is sensitive to the site position of the WP in the ground state potential. It is clear that the dynamics of the center of gravity of the X-ray spectrum follows one-to-one to the trajectory of the WP. In the following we investigate in detail the revival in the OK X-ray absorption of the NO molecule.
3. Computational details We start with a brief outline of the computational details described in more detail earlier [27,28]. The simulations are performed for OK photoabsorption in the nitrogen monoxide molecule, NO. We consider here the most intense low energy electronic transition: NO(2P) ! NO*(2R). The propagation of the WPs is simulated using Morse potentials of the ground and core excited states from Ref. [29]. The lifetime broadening of the core excited state (C 0.08 eV) of the NO molecule is neglected, as well as the broadening (0.01 eV) due to the spectral function of the X-ray field . These approximations are quite reasonable due to the large broadening of the spectral profile, 1/sX 0.16 eV, caused by the short X-ray pulse with a half-width at half-maximum (HWHM), sX = 4 fs. Both IR and X-ray pulses are modeled by Gaussians. The simulations are divided in two parts. In the first step, we compute the nuclear WP /(t) in the ground electronic state, NO(2P), solving numerically the Schro¨dinger equation (2) without any assumption about the intensity of the IR field. In this step the IR field is assumed to be in resonance with the first vibrational transition, xL = x10 and the small lifetime broadenings of the vibrational states as well as the collisional dephasing are neglected. The peak position, the duration and peak intensity of the IR pulse are tL = 700 fs, sL = 100 fs, and IL = 2.3 · 1012 W/cm2, respectively. Such an IR pulse creates a coherent superposition of vibrational states with the populations: q00 = 0.208, q11 = 0.605, q22 = 0.182 and q33 = 0.005. Apparently, the peak position of the IR pulse tL = 700 fs does not influence the pump–probe spectrum, which is sensitive only to the delay time between the pump and the probe pulses. The second step consists of the evaluation of the nuclear wave packet /c(t) in the potential of the core excited state NO*(2R) and the Fourier transform /c(X)
F.F. Guimara˜es et al. / Chemical Physics Letters 405 (2005) 398–403
ð9Þ
1
where H0 is the nuclear Hamiltonian of the core-excited state, Dc0 is the transition dipole moment of core-excitation. The probability of X-ray absorption P(X) is calculated as the norm of the WP in the frequency domain [27,28] P ðXÞ ¼ h/c ðXÞj/c ðXÞi:
ð10Þ
The detuning of the X-ray field X = x xc0 is defined relative to the adiabatic excitation energy xc0 = 531.3 eV. The initial wave function j0æ and wave packets /(t), /c(t), are calculated employing, respectively, time independent and time dependent techniques [30], implemented in the eSPec program [31]. All the simulations are performed for a fixed in space molecule with the molecular axis being parallel to the polarization vector of the IR field eL. One can accomplish this orientation by detection of X-ray absorption in the ion yield mode [32].
4. Trajectories of wave packets and X-ray absorption profiles
2
the WPs DrðtÞ ¼ hr2 ðtÞi hrðtÞi (see Fig. 3a and b). The regions with largest amplitude of oscillations of Ær(t)æ or ÆXæ correspond to a highly coherent quasiclassical behavior with a well-localized WP (small Dr(t)). The anharmonicity results in quantum mechanical dephasing (suppression of amplitude of oscillations of Ær(t)æ) and the WP spreads. After the revival time, Trev = 1210 fs, the coherence is restored and the WP localizes again. Our calculations show that both the WP and the X-ray spectrum restore the shape through the revival period and q that the width of the X-ray spectrum ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 DX ¼ hX i hXi2 has almost the same time dependence as Dr(t). The important parameter of revival is the visibility or contrast of the revival modulation. Strict calculations show that the contrast of the revival modulations (Ær(t)æmax re)/(Ær(t)æmin re) 14 is two times larger than the value obtained from Eq. (8). The reason for this is that Eq. (8) is valid only for f 1, while for the used
0.09 0.06
(b)
〈Ω〉 (eV) Fig. 2. Trajectory of the WP and the center gravity of the OK X-ray spectrum of NO (solid lines). Both trajectories coincide with high precision (the only difference is the scales of Ær(t)æand ÆXæ). The right panel displays the phase sensitivity of the trajectory. Filled and dashed bands at the right-hand side display, respectively, the WP and the X-ray spectrum. Dt = t tL and Dt = tX tL for Ær(t)æand ÆXæ, respectively. sX = 4 fs.
0.15 0.12 (a)
1.2 0.9 0.6 0.3
〈Ω〉 (eV)
∆r (a.u.)
Fig. 2 displays the propagation of the center of gravity of the WPs and of the X-ray absorption profiles R XP ðXÞ dX hrðtÞi ¼ h/ðtÞjrj/ðtÞi; hXi ¼ R ; ð11Þ P ðXÞ dX
where ÆXæ depends on the delay time between the X-ray and IR pulses, Dt = tX tL. The simulations show that the two trajectories coincide with high precision. Such a coincidence prevails because the wave packet /(t) propagates (for IL = 2.3 · 1012 W/cm2) in the region where the slope of the core excited potential is negative, dEC(R)/dR < 0 (Fig. 1). For higher intensity of the IR pulse the WP moves in the region with positive slope dEC(R)/dR > 0. In this case the trajectory of the spectrum ÆXæceases to copy the trajectory of the WP, Ær(t)æ. One can see that the trajectories of both the wave packet and the X-ray spectrum display the revival phenomenon with the revival time Trev 1210 fs. It is relevant to note that these trajectories are very sensitive to the phase of the IR field uL in a short time scale [25–28] (see righthand side panel of Fig. 2). However, the large time scale modulations caused by the revival phenomenon does not depend on the phase uL. It is instructive to compare the trajectory of the WP Ær(t)æ or of the X-ray spectrum ÆXæ ffi with the spread of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1.2 (c) 0.9 0.6 0.3 500
Long pulses Short pulses
1 j/c ðtÞi ¼ eiH c t fj/ðtÞi; f ¼ ðeX Dc0 Þ; 2 Z 1 iXt dte EX ðtÞj/c ðtÞi; j/c ðXÞi ¼
401
1000
1500
2000
2500
t (fs) Fig. 3. The spread of the WP Dr(t) (panel a) versus the center of gravity of the X-ray spectrum ÆXæ (panels b and c). uL = p. (b) Short X-ray pulse, sX = 4 fs. (c) Long X-ray pulse, sX = 30 fs.
402
F.F. Guimara˜es et al. / Chemical Physics Letters 405 (2005) 398–403 −4
-1
−4
-1
γ = 1.0 × 10 fs
2.3 2.2
hence, the intermolecular interaction, making use of X-ray spectroscopy.
2.1 〈r〉 (a. u.)
2.3
= 5.0 × 10 fs
2.2
5. Conclusions
2.1 −3
2.3
-1
= 1.0 × 10 fs
2.2 2.1 1000
2000
3000
4000
5000
t (fs) Fig. 4. Damping of the revival structure due to dephasing collisions. sX = 4 fs.
intensity IL = 2.3 · 1012 W/cm2 the parameter f (6) is rather large, f 1.3. The comparison of Figs. 3b and c shows that the revival phenomenon can be observed only for the duration of X-ray pulses shorter (sX = 4 fs) than the period of vibrations T = 2p/x10 17 fs. This means that a smaller vibrational frequency (heavier molecule) allows to use a longer X-ray pulse, which is desirable from the experimental point of view. The center of gravity of the X-ray spectrum does almost not depend on the delay time when the X-ray pulse is longer than T (sX = 30 fs). The reason for this is that a long X-ray pulse gives a spectrum integrated over a large time domain, which means that the fast quasiclassical oscillations are diminished. Until now we analyzed results of simulations neglecting the decay of the vibrational coherence. This is justified for low temperatures when only the lowest rotational level, J = 0, is populated and for low pressure with negligible collisional dephasing (c = 0). However, the revival phenomenon is caused by the coherence between adjacent vibrational levels and is therefore very sensitive to decoherence induced by rotations and collisions with buffer molecules [14,16,17]. The rotational dephasing can be considerably suppressed for surface adsorbate molecules or for molecules embedded in a solid matrix. To point out the importance of the longterm structure in the X-ray probe signal we consider here only the collisional dephasing which is defined by the rate constant c ¼ vrN , where v and r are the thermal velocity, respectively, the cross section of dephasing collisions between the NO molecule and buffer particles with the concentration N. Eq. (5) says that the trajectory of the wave packet as well as of the X-ray spectrum can be written as [17] Ær(t)æ = Ær(t)æN=0 · exp(ct 0 ). The simulations (Fig. 4) show that the pump–probe spectra are very sensitive to the dephasing rate c which is proportional to the gas pressure. This gives an opportunity to measure the cross section of dephasing collisions and,
The recent progress in development of ultrashort X-ray pulses [33–35] has brought entirely new possibilities into practice for time-resolved X-ray spectroscopy. In this work we have suggested and explored a promising scheme for a X-ray pump–probe experiment taking advantage of short X-ray pulses. We have investigated the revival effect in IR–X-ray pump–probe spectra caused by the anharmonicity of the molecular potential. This phenomenon can be seen for X-ray pulses shorter than the period of nuclear vibrations. It is shown that the short-term dynamics of the spectrum is sensitive to the phase of the IR field contrary to the long-term revival structure which is phase independent. The observation of slow dynamics of the center of gravity of the X-ray probe spectrum enables to measure the revival period and, hence, the anharmonicity constant. This is beyond the possibilities of ordinary X-ray absorption spectroscopy with the resolution restricted by the lifetime broadening of core excited state (for the studied molecule x10xe = 0.0017 eV and C 0.08 eV). Here, we thus face a somewhat paradoxial situation in that the short X-ray pulse allows to measure the fine structure of the molecular spectra. Indeed, the idea of ultrashort pulses seems insurmountable for high precision spectroscopy; as we make pulses shorter and shorter, we enlarge the pulse bandwidth, and loose spectral resolution. As a matter of fact, there is here no contradiction with the uncertainty relation because the anharmonicity is determined via long time measurements. Another important application of the revival phenomenon in X-ray spectroscopy is the detection of the slow vibrational coherence decay in the ground electronic state with help of short X-ray pulses. Such longterm measurements enable to study the intermolecular interaction. We also point out that IR–X-ray pump– probe spectroscopy can be a promising tool in structure studies of liquids. Due to vibrational selectivity the IR pulse can excite certain structures of the liquid which subsequently can be snapshoted by the X-ray pulse. The IR induced X-ray absorption of certain structures can be extracted from the total X-ray spectrum by means of a modulation of the IR intensity or using the revival effect which is also structure sensitive. Already now the X-ray pulses shorter than 1 fs are available [34], something that makes possible the observation of the revival effect in X-ray spectra. For longer X-ray pulses (100 fs) we have to study heavier molecules with smaller vibrational frequencies.
F.F. Guimara˜es et al. / Chemical Physics Letters 405 (2005) 398–403
Acknowledgments This work was supported by the Swedish Research Council (VR) and by the STINT foundation. F.F.G. acknowledge financial support from Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico (CNPq – Brazil). F.G acknowledge also financial support from the Russian Foundation for Basic Research (Project No. 04-02-81020-Bel2004).
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Quantum wave packet revivals in IR + X-ray pump–probe spectroscopy F.F. Guimara˜es
˚ gren , F. Gelmukhanov a, A. Cesar b, H. A
a,b,*
a
a
b
Theoretical Chemistry, Roslagstullsbacken 15, Royal Institute of Technology, S-106 91 Stockholm, Sweden Departamento de Quı´mica, Universidade Federal de Minas Gerais, Av. Antonio Carlos 6627, CEP-31270-901, Belo Horizonte, Minas Gerais, Brazil Received 10 January 2005; in final form 14 February 2005 Available online 16 March 2005
Abstract The wave packet revivals constitute a central concept of X-ray spectroscopy with ultra-high spectral resolution. The revival phenomenon allows to resolve the anharmonical shift or rotational structure by means of time dependent measurements and makes X-ray pump–probe spectroscopy a powerful technique to study long-term dynamics of molecules in different phases. We study the revivals referring to the X-ray absorption spectrum of the NO molecule driven by strong infrared pulse. It is shown that the phase sensitive trajectories of the center of gravity of the wave packets and the X-ray spectra copy each other. 2005 Elsevier B.V. All rights reserved.
1. Introduction Advances in the physics and chemistry of laser interactions with atoms and molecules have brought the concept of wave-packets and their dynamics into the limelight. One of the major reasons for studying wave packet dynamics in the context of molecules is related to laser catalysis [1,2] and the control of chemical reactions [3–8] by careful application of pulses of light with optimal frequency, intensity, duration, and timing. Among these studies a special attention has been paid to the long-term dynamics related to quantum wave packet revivals. Classical as well as quantum dynamics of systems of coupled oscillators with slightly different frequencies experience fast oscillations with vibrational frequency and slow modulations. These modulations, defined by the differences between the frequencies of individual oscillators, are named revivals. Revival phenomena are *
Corresponding author. Fax: +46 8 5537 8590. E-mail addresses: [email protected], [email protected] (F.F. Guimara˜es). 0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.02.061
observed in many branches of physics. The first studies related to the revival time are traced to the Poincare recurrences for a rotation map [9], where the revival is analyzed in classical systems. Quantum revivals are usually associated with the dynamics of wave packets (WP) invented in physics by Schro¨dinger [10]. Quantum mechanical dephasing caused by anharmonicity (nonequidistant spectrum) leads to a delocalization of the initial wave packet which regains its initial shape after the revival period. The wave packet revival, predicted by Parker and Stroud [11], was experimentally confirmed a few years later [12]. Wave packet revivals are recognized to be important, for instance, in the motion of Rydberg wave packets [11,13], for rotational and vibrational degrees of freedom [14–17], in core-excited states [18,19], for pulse shaping [20,21], and for isotope separation [22]. The current state of theoretical and experimental studies of the wave packet revivals in atomic, molecular, condensed matter and optical systems has been reviewed by Robinett [23] a short time ago. The quantum revival is essentially a coherent phenomenon related to the dynamics of coherent superposition of quantum states. The wave packets can be
2. Physical picture of quantum revivals in an X-ray absorption spectrum We consider molecules that interact with IR pump field (L) and probe X-ray radiation (X) (see Fig. 1): Ea ðtÞ ¼ Ea ðtÞ cosðxa t ka R þ ua Þ;
a ¼ L; X :
ð1Þ
The pump and probe fields Ea(t) = eaEa(t) are characterized by the polarization ea, the wave vectors envelopes ka, the frequencies xa and the phases ua. Atomic units are assumed everywhere in this Letter. The IR radiation influences the X-ray absorption in two different ways: due to the population of vibrational levels (m) and through the IR induced coherence between these levels. The coherent IR light mixes different vibrational states and creates their coherent superposition or the wave packet. Since the wave packet is not an eigenfunction of the ground state Hamiltonian, it starts to move. The coherently created wave packet performs the back and forth propagation in the potential well of the ground electronic state. X-ray snapshots of the nuclear
534.0
Energy (eV)
generated in the potential energy surface due to optical [14] or X-ray transitions [24] as well as by means of vibrational–rotational transitions in the field of an infrared (IR) laser. The coherent properties of wave packets induced by an IR field are related directly to their phases and gives the opportunity to construct different superpositions of quantum states which are sensitive to the phase of the pump radiation. This constitutes the background of phase sensitive IR–X-ray pump–probe spectroscopy [25–28]. The main aim of our article is to study the role of quantum revivals in IR–X-ray pump–probe spectroscopy. Contrary to optical and IR spectroscopies, conventional X-ray spectroscopy has rather poor spectral resolution, mainly due to the rather large lifetime broadening and because of the quite large instrumental broadening. We show that the center of gravity of X-ray probe spectrum experiences modulation in the time domain with the revival period inversely proportional to the anharmonicity constant which is smaller than the lifetime broadening of X-ray resonance. Thus, the revival phenomenon makes this pump–probe setup very promising in X-ray spectroscopy with ultra-high resolution, far beyond the limitations due to the lifetime broadening. Measurements of the revival period give direct information about hyperfine structure of the molecular spectrum. We will also show how the X-ray pump–probe spectroscopy can be used in studies of inter- and intramolecular interactions. We deal with revivals related to the anharmonicity of the interatomic potential in the ground electronic state.
399
Ω (eV)
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533.0
A
532.0
B
531.0 Absorption probability
5.0 4.0 3.0 2.0 1.0 0.0 1.5
A
2.0
B
2.5
3.0
3.5
r (a.u.) Fig. 1. Scheme of OK X-ray absorption of NO driven by a strong IR field. At a certain instant a short X-ray pulse promotes the wave packet (A or B) to a certain point of the core excited potential. The X-ray absorption profiles A and B show the dependence of the X-ray spectra on the delay between X-ray and IR pulses.
wave packet at different site positions yield the X-ray spectra (see Fig. 1). This technique maps the trajectory of the WP and is useful in studies of intramolecular interaction in excited states [28,27]. The ground-states nuclear wave packet obeys the Schro¨dinger equation i
o ^ þ C /ðtÞ ¼ ½H 0 ðd EL ðtÞÞ cosðxL t þ uL Þ/ðtÞ; ot ð2Þ
^ and with the Hamiltonian H0, the relaxation matrix C initial condition j/(0)æ = j0æ. We neglect the spatial phases kL Æ R and kX Æ R the role of which was already discussed [25]. Before embarking on the details of the calculations, we employ simple arguments based on the rotating wave approximation (xL x10): when the IR pulse leaves the system the dynamics of the wave packet (2) is straightforwardly described by X jmiam eðim þCm Þðtt0 Þ ; j/ðtÞi ¼ m
am ¼ i e
iuL
Z
1
dt1 ðdm;m1 EL ðt1 ÞÞ 1 e½iðxm;m1 xL ÞþCm Cm1 ðt1 t0 Þ am1 ðt1 Þ
¼ jam jeiðmuL þvm Þ ; where vm is the intrinsic phase of the vibrational state m, which depends on the frequency and the shape of the IR pulse; t0 is the time when the IR pulse leaves the system, xm,m1 = m m1; m and jmæ are the vibrational energy and eigenvector of the ground electronic state, and; Cm is the decay rate of the vibrational level m (C0 = 0). It can be seen that even after that the IR pulse
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left the system the wave packet continues to be coherent and keeps the memory about the laser phase uL during the lifetime of the vibrational levels, C1 m . The physical understanding of the revival phenomenon can be achieved by considering the IR field of moderate intensity which mixes only the first three vibrational levels, m = 0,1,2. It is instructive to analyze the trajectory of the wave packet which is characterized by its center of gravity hrðtÞi ¼ h/ðtÞjrj/ðtÞi ¼ re þ 2Re q10 eðix10 þC10 Þðtt0 Þ r01 þq21 eðix21 þC21 Þðtt0 Þ r12 :
ð3Þ
Here, qmm1 ¼ am am1 is the density matrix, rmm1 ¼ hmjrjm1 i, re is the equilibrium interatomic separation, and Cmm1 ¼ Cm þ Cm1 . When the molecule is embedded in a bath (gas or liquid), it experiences collisions which quench the coherence qm,m1 with the rate cm,m1 c. Such a dephasing increases the decay rate of the coherence Cm;m1 ¼ Cm þ Cm1 þ c:
ð4Þ
To be specific, we assume that the density of the sample is quite high: Cm,m1 c. The resulting expression for the center of gravity (3) can be written as follows pffiffiffiffiffiffiffiffiffiffiffiffiffi hrðtÞi ¼ re þ 2r01 q00 q11 ð1 fÞ cosðx10 t0 þ u10 Þ t0 0 þ 2f cosðx10 t0 þ uþ Þ cos 2p þ u ect ; TR ð5Þ 0
where t = t t0 P 0, u± = (u10 ± u21)/2, u10 = uL + u1 u0, u21 = uL + u2 u1. We used here the harmonic approximation pffiffifor ffi the ratio of transition dipole moments, r12 =r01 ¼ 2, and introduced the auxiliary parameter sffiffiffiffiffiffiffiffiffi 2q22 : ð6Þ f¼ q00 The center of gravity of the wave packet (5) experiences fast oscillations with the period inversely proportional to the vibrational frequency x10 which are modulated by slow oscillations with the period equal to the revival time TR ¼
4p 2p ¼ x10 x21 x10 xe
ð7Þ
inversely proportional to the anharmonicity constant x10 xe = 13.71 cm1 = 0.0017 eV. This equation follows from the expression for the eigenvalues of the Morse oscillator: m = x10(m + 1/2)x10xe(m + 1/2)2. The important characteristic of the WP revival is its contrast, (Ær(t)æmax re)/jÆr(t)æmin rej. When the population of the vibrational level m = 2 is small the contrast is related directly to the parameter f (6)
hrðtÞimax re 1þf hrðtÞi r ¼ j1 fj ; e min
f 1:
ð8Þ
Our simulations show that for large f this ratio becomes sensitive to the phases u10 and u21. The wave packet revivals can be observed making use of X-ray absorption as is illustrated in Fig. 1. The potentials of the ground and core-excited states usually differ, which implies that the X-ray absorption is sensitive to the site position of the WP in the ground state potential. It is clear that the dynamics of the center of gravity of the X-ray spectrum follows one-to-one to the trajectory of the WP. In the following we investigate in detail the revival in the OK X-ray absorption of the NO molecule.
3. Computational details We start with a brief outline of the computational details described in more detail earlier [27,28]. The simulations are performed for OK photoabsorption in the nitrogen monoxide molecule, NO. We consider here the most intense low energy electronic transition: NO(2P) ! NO*(2R). The propagation of the WPs is simulated using Morse potentials of the ground and core excited states from Ref. [29]. The lifetime broadening of the core excited state (C 0.08 eV) of the NO molecule is neglected, as well as the broadening (0.01 eV) due to the spectral function of the X-ray field . These approximations are quite reasonable due to the large broadening of the spectral profile, 1/sX 0.16 eV, caused by the short X-ray pulse with a half-width at half-maximum (HWHM), sX = 4 fs. Both IR and X-ray pulses are modeled by Gaussians. The simulations are divided in two parts. In the first step, we compute the nuclear WP /(t) in the ground electronic state, NO(2P), solving numerically the Schro¨dinger equation (2) without any assumption about the intensity of the IR field. In this step the IR field is assumed to be in resonance with the first vibrational transition, xL = x10 and the small lifetime broadenings of the vibrational states as well as the collisional dephasing are neglected. The peak position, the duration and peak intensity of the IR pulse are tL = 700 fs, sL = 100 fs, and IL = 2.3 · 1012 W/cm2, respectively. Such an IR pulse creates a coherent superposition of vibrational states with the populations: q00 = 0.208, q11 = 0.605, q22 = 0.182 and q33 = 0.005. Apparently, the peak position of the IR pulse tL = 700 fs does not influence the pump–probe spectrum, which is sensitive only to the delay time between the pump and the probe pulses. The second step consists of the evaluation of the nuclear wave packet /c(t) in the potential of the core excited state NO*(2R) and the Fourier transform /c(X)
F.F. Guimara˜es et al. / Chemical Physics Letters 405 (2005) 398–403
ð9Þ
1
where H0 is the nuclear Hamiltonian of the core-excited state, Dc0 is the transition dipole moment of core-excitation. The probability of X-ray absorption P(X) is calculated as the norm of the WP in the frequency domain [27,28] P ðXÞ ¼ h/c ðXÞj/c ðXÞi:
ð10Þ
The detuning of the X-ray field X = x xc0 is defined relative to the adiabatic excitation energy xc0 = 531.3 eV. The initial wave function j0æ and wave packets /(t), /c(t), are calculated employing, respectively, time independent and time dependent techniques [30], implemented in the eSPec program [31]. All the simulations are performed for a fixed in space molecule with the molecular axis being parallel to the polarization vector of the IR field eL. One can accomplish this orientation by detection of X-ray absorption in the ion yield mode [32].
4. Trajectories of wave packets and X-ray absorption profiles
2
the WPs DrðtÞ ¼ hr2 ðtÞi hrðtÞi (see Fig. 3a and b). The regions with largest amplitude of oscillations of Ær(t)æ or ÆXæ correspond to a highly coherent quasiclassical behavior with a well-localized WP (small Dr(t)). The anharmonicity results in quantum mechanical dephasing (suppression of amplitude of oscillations of Ær(t)æ) and the WP spreads. After the revival time, Trev = 1210 fs, the coherence is restored and the WP localizes again. Our calculations show that both the WP and the X-ray spectrum restore the shape through the revival period and q that the width of the X-ray spectrum ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 DX ¼ hX i hXi2 has almost the same time dependence as Dr(t). The important parameter of revival is the visibility or contrast of the revival modulation. Strict calculations show that the contrast of the revival modulations (Ær(t)æmax re)/(Ær(t)æmin re) 14 is two times larger than the value obtained from Eq. (8). The reason for this is that Eq. (8) is valid only for f 1, while for the used
0.09 0.06
(b)
〈Ω〉 (eV) Fig. 2. Trajectory of the WP and the center gravity of the OK X-ray spectrum of NO (solid lines). Both trajectories coincide with high precision (the only difference is the scales of Ær(t)æand ÆXæ). The right panel displays the phase sensitivity of the trajectory. Filled and dashed bands at the right-hand side display, respectively, the WP and the X-ray spectrum. Dt = t tL and Dt = tX tL for Ær(t)æand ÆXæ, respectively. sX = 4 fs.
0.15 0.12 (a)
1.2 0.9 0.6 0.3
〈Ω〉 (eV)
∆r (a.u.)
Fig. 2 displays the propagation of the center of gravity of the WPs and of the X-ray absorption profiles R XP ðXÞ dX hrðtÞi ¼ h/ðtÞjrj/ðtÞi; hXi ¼ R ; ð11Þ P ðXÞ dX
where ÆXæ depends on the delay time between the X-ray and IR pulses, Dt = tX tL. The simulations show that the two trajectories coincide with high precision. Such a coincidence prevails because the wave packet /(t) propagates (for IL = 2.3 · 1012 W/cm2) in the region where the slope of the core excited potential is negative, dEC(R)/dR < 0 (Fig. 1). For higher intensity of the IR pulse the WP moves in the region with positive slope dEC(R)/dR > 0. In this case the trajectory of the spectrum ÆXæceases to copy the trajectory of the WP, Ær(t)æ. One can see that the trajectories of both the wave packet and the X-ray spectrum display the revival phenomenon with the revival time Trev 1210 fs. It is relevant to note that these trajectories are very sensitive to the phase of the IR field uL in a short time scale [25–28] (see righthand side panel of Fig. 2). However, the large time scale modulations caused by the revival phenomenon does not depend on the phase uL. It is instructive to compare the trajectory of the WP Ær(t)æ or of the X-ray spectrum ÆXæ ffi with the spread of qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1.2 (c) 0.9 0.6 0.3 500
Long pulses Short pulses
1 j/c ðtÞi ¼ eiH c t fj/ðtÞi; f ¼ ðeX Dc0 Þ; 2 Z 1 iXt dte EX ðtÞj/c ðtÞi; j/c ðXÞi ¼
401
1000
1500
2000
2500
t (fs) Fig. 3. The spread of the WP Dr(t) (panel a) versus the center of gravity of the X-ray spectrum ÆXæ (panels b and c). uL = p. (b) Short X-ray pulse, sX = 4 fs. (c) Long X-ray pulse, sX = 30 fs.
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-1
−4
-1
γ = 1.0 × 10 fs
2.3 2.2
hence, the intermolecular interaction, making use of X-ray spectroscopy.
2.1 〈r〉 (a. u.)
2.3
= 5.0 × 10 fs
2.2
5. Conclusions
2.1 −3
2.3
-1
= 1.0 × 10 fs
2.2 2.1 1000
2000
3000
4000
5000
t (fs) Fig. 4. Damping of the revival structure due to dephasing collisions. sX = 4 fs.
intensity IL = 2.3 · 1012 W/cm2 the parameter f (6) is rather large, f 1.3. The comparison of Figs. 3b and c shows that the revival phenomenon can be observed only for the duration of X-ray pulses shorter (sX = 4 fs) than the period of vibrations T = 2p/x10 17 fs. This means that a smaller vibrational frequency (heavier molecule) allows to use a longer X-ray pulse, which is desirable from the experimental point of view. The center of gravity of the X-ray spectrum does almost not depend on the delay time when the X-ray pulse is longer than T (sX = 30 fs). The reason for this is that a long X-ray pulse gives a spectrum integrated over a large time domain, which means that the fast quasiclassical oscillations are diminished. Until now we analyzed results of simulations neglecting the decay of the vibrational coherence. This is justified for low temperatures when only the lowest rotational level, J = 0, is populated and for low pressure with negligible collisional dephasing (c = 0). However, the revival phenomenon is caused by the coherence between adjacent vibrational levels and is therefore very sensitive to decoherence induced by rotations and collisions with buffer molecules [14,16,17]. The rotational dephasing can be considerably suppressed for surface adsorbate molecules or for molecules embedded in a solid matrix. To point out the importance of the longterm structure in the X-ray probe signal we consider here only the collisional dephasing which is defined by the rate constant c ¼ vrN , where v and r are the thermal velocity, respectively, the cross section of dephasing collisions between the NO molecule and buffer particles with the concentration N. Eq. (5) says that the trajectory of the wave packet as well as of the X-ray spectrum can be written as [17] Ær(t)æ = Ær(t)æN=0 · exp(ct 0 ). The simulations (Fig. 4) show that the pump–probe spectra are very sensitive to the dephasing rate c which is proportional to the gas pressure. This gives an opportunity to measure the cross section of dephasing collisions and,
The recent progress in development of ultrashort X-ray pulses [33–35] has brought entirely new possibilities into practice for time-resolved X-ray spectroscopy. In this work we have suggested and explored a promising scheme for a X-ray pump–probe experiment taking advantage of short X-ray pulses. We have investigated the revival effect in IR–X-ray pump–probe spectra caused by the anharmonicity of the molecular potential. This phenomenon can be seen for X-ray pulses shorter than the period of nuclear vibrations. It is shown that the short-term dynamics of the spectrum is sensitive to the phase of the IR field contrary to the long-term revival structure which is phase independent. The observation of slow dynamics of the center of gravity of the X-ray probe spectrum enables to measure the revival period and, hence, the anharmonicity constant. This is beyond the possibilities of ordinary X-ray absorption spectroscopy with the resolution restricted by the lifetime broadening of core excited state (for the studied molecule x10xe = 0.0017 eV and C 0.08 eV). Here, we thus face a somewhat paradoxial situation in that the short X-ray pulse allows to measure the fine structure of the molecular spectra. Indeed, the idea of ultrashort pulses seems insurmountable for high precision spectroscopy; as we make pulses shorter and shorter, we enlarge the pulse bandwidth, and loose spectral resolution. As a matter of fact, there is here no contradiction with the uncertainty relation because the anharmonicity is determined via long time measurements. Another important application of the revival phenomenon in X-ray spectroscopy is the detection of the slow vibrational coherence decay in the ground electronic state with help of short X-ray pulses. Such longterm measurements enable to study the intermolecular interaction. We also point out that IR–X-ray pump– probe spectroscopy can be a promising tool in structure studies of liquids. Due to vibrational selectivity the IR pulse can excite certain structures of the liquid which subsequently can be snapshoted by the X-ray pulse. The IR induced X-ray absorption of certain structures can be extracted from the total X-ray spectrum by means of a modulation of the IR intensity or using the revival effect which is also structure sensitive. Already now the X-ray pulses shorter than 1 fs are available [34], something that makes possible the observation of the revival effect in X-ray spectra. For longer X-ray pulses (100 fs) we have to study heavier molecules with smaller vibrational frequencies.
F.F. Guimara˜es et al. / Chemical Physics Letters 405 (2005) 398–403
Acknowledgments This work was supported by the Swedish Research Council (VR) and by the STINT foundation. F.F.G. acknowledge financial support from Conselho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico (CNPq – Brazil). F.G acknowledge also financial support from the Russian Foundation for Basic Research (Project No. 04-02-81020-Bel2004).
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