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Phase control effects induced by a detuning of third laser harmonic from triple fundamental frequency
Chemical Physics Letters 409 (2005) 8–13 www.elsevier.com/locate/cplett
Phase control effects induced by a detuning of third laser harmonic from triple fundamental frequency Alexander I. Pegarkov
*,1
Chemical Physics Theory Group, Department of Chemistry, University of Toronto, 80 St. George Street, Toronto, Ont., Canada M5S 3H6 Received 7 February 2005; in final form 5 April 2005 Available online 23 May 2005
Abstract The phase control of the absorption of three photons from the fundamental laser mode and one photon from its third harmonic is studied allowing for a small difference between frequency of the third harmonic and the triple frequency of the fundamental wave. The effect of triple wave detuning upon the (x1, x3) absorption is illustrated numerically for model molecule NH3. It is shown that in experiments the phase control can be affected by such a frequency detuning. Several experimental schemes are discussed in order to check out and to confirm this effect. 2005 Elsevier B.V. All rights reserved.
1. Introduction More than 15 years ago, the first theoretical study [1] showed that the bichromatic excitation provided by the fundamental laser line and its third harmonic could afford an experimentally simple route to control over unimolecular reactions. Since that time the experimental methods have been excellently developed and permit now to control the (x1, 3x1) excitation and ionization of simple and large polyatomic molecules by a change of either relative phase or intensity of fundamental and third laser harmonics [2–6] . In last decade due to a fast development in the technique of short-pulse lasers the strong experimental and theoretical efforts have been making also to use the femtosecond pulses of powerful lasers in order to
*
Fax: +1 613 562 5190. E-mail address: [email protected]. 1 Present address (since January 1, 2004): Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Ont., Canada K1N 6N5. 0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.04.089
achieve effective control over photochemical processes [7–10]. In contrast to the laser beam experiments [2–6], a linear dispersion of the frequency-multiplying medium, and a frequency chirp create also some mismatch between frequency of the nÕs harmonic and n multiplied the fundamental laser frequency, which can change the output of a chemical process controlled by pulsed laser. In experiments using pulsed lasers one gets the third harmonic of the frequency slightly different from the triple fundamental frequency [11,12]: x3 = 3x1 + dx, Fig. 1. Actually the detuning of a triple wave dx is very small like (106 104)x1 and was never taken into account in theoretical and experimental analysis. The goal of this Letter is to treat the one-photon versus three-photon control scheme in the case of triple wave detuning: (x1, 3x1 + dx). The results demonstrate that, although the frequency detuning is small, it visibly impacts upon the phase control and changes the chemical product output. The inclusion of triple wave detuning into the quantum control theory [13] can help to explain an unusual modulation depth behavior observed earlier in experiments [5,6].
A.I. Pegarkov / Chemical Physics Letters 409 (2005) 8–13
9
where H0 is the unperturbed molecular Hamiltonian, HD is a phenomenological anti-hermitian operator causing a laser-free decay to the bath, V1(t) and V3(t) are the operators of dipole interaction of the molecule with the fundamental laser wave and its third harmonic V 1;3 ðtÞ ¼ ðd; E1;3 Þf ðs; tÞ cosðx1;3 t þ /1;3 Þ; d is the molecular dipole moment and f(s, t) is the pulse shape function, 1; 0 6 t 6 s; f ðs; tÞ ¼ 0; t < 0; t > s.
Fig. 1. Excitation diagram for phase control with two interfering amplitudes created by three-photon and one-photon transitions from the ground molecular state. (a) The ideal case, where the frequency of the third harmonic, x3 is sharply equal to the triple frequency of the fundamental laser line, x3 = 3x1. (b) A realistic case, where x3 and 3x1 are slightly detuned, x3 = 3x1 + dx.
Let us consider a two-level quantum system irradiated by a laser wave x1 and its third harmonic x3 having a frequency detuning (Fig. 1b) EðtÞ ¼ E1 cosðx1 t þ /1 Þ þ E3 cosðx3 t þ /3 Þ; dx ¼ kx1
and assume that the initial molecular state is the ground electronic-rotation–vibration state while the final state is an excited one having an energy width due to various non-radiative transitions to a bath of other excited molecular states and continua Ej ¼ j iCj =2. In polyatomic molecules these transitions are much faster than spontaneous decay to the ground state, which is not included here. The quantum system, been excited to state |jæ, can then absorb two photons from fundamental mode and reach ionization continuum. In experiment the dynamics of resonance enhanced multiphoton ionization repeats the population dynamics of excited molecular state. This model therefore does not include ionization continuum and assumes that the direct laser-induced transitions from both states to the bath are forbidden as well. The temporal dynamics of the one photon versus three photon phase control is described by the following time-dependent Schro¨dinger equation oWðtÞ ¼ H ðtÞWðtÞ; ot H ðtÞ ¼ H 0 þ H D þ V 1 ðtÞ þ V 3 ðtÞ;
ih
U0 ðtÞ ¼ eix0 t j/0 i; Uj ðtÞ ¼ eiðxj icj =2Þt j/j i;
ð2Þ
x0 ¼ E0 =h; xj ¼ j =h; cj ¼ Cj =h; where the functions |/0æ, |/jæ fit the following time-independent Schro¨dinger equations
2. Coherent control theory for the detuned (x1, x3) absorption to a decaying quantum state
x3 ¼ 3x1 þ dx;
The laser-free states of our two-level model molecule are the quasi-stationary states which obey the wave equation like Eq. (1) having a time-independent laserfree Hamiltonian. The laser-free quasi-stationary states can be written as
ð1Þ
½H 0 þ H D j/0 i ¼ E0 j/0 i; Cj ½H 0 þ H D j/j i ¼ j i j/j i. 2
ð3Þ ð4Þ
Before the laser pulse reaches the model molecule, the molecule is in its ground state. As the laser field begins to interact with the molecule, then both molecular states are populated and the population maximum oscillates between both molecular states. Once two laser modes have different frequencies, the population oscillation frequency differs from the Rabi frequencies independently induced by each laser mode and the full population dynamics displays a complex pattern created by two time-dependent perturbations. At a time t during the laser–molecule interaction, 0 6 t 6 Dtint, the total time-dependent state can be represented by the following superposition WðtÞ ¼ b0 ðtÞj/0 i þ bj ðtÞj/j i.
ð5Þ
Substituting Eq. (5) into Eq. (1), multiplying it by the proper bra- state vectors and having in mind Eqs. (3) and (4), we obtain the following matrix equation for the time-dependent state amplitudes b0(t) and bj(t) d BðtÞ ¼ HðtÞBðtÞ; dt B(t) is the state amplitude vector b0 ðtÞ BðtÞ ¼ bj ðtÞ
ð6Þ
ð7Þ
and H(t) is the following non-diagonal symmetric matrix
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A.I. Pegarkov / Chemical Physics Letters 409 (2005) 8–13
HðtÞ ¼
ix0 2i½V 1 cosðx1 t þ /1 Þ þ V 3 cosðx3 t þ /3 Þ
The initial condition for Eq. (6) is 1 Bðt ¼ 0Þ ¼ . 0
ð9Þ
Finally, Eqs. (6)–(9) present a phase control version of the time-dependent Schro¨dinger equation for a two-level system with decay [14].
3. Phase modulation of the detuned (x1, x3) absorption: numerical results and discussion Let us show that the time-dependent quantum interference of the slightly incoherent x3 and 3x1 pathways significantly impacts upon the phase control and can be a possible source of unusual phase modulation observed experimentally in [5,6]. In [5,6] the ion signal of two-photon ionization from the excited |jæ state has been measured. The ions are produced in the point of intersection of the laser beam and molecular beam. Therefore, the ion signal generated by this way is proportional to the |j æ |0æ bichromatic absorption probability Pj(/, t) integrated over the time of the laser–molecule interaction Z Dtint ið/Þ P j ð/; tÞ dt; ð10Þ 0
P j ð/; tÞ jh/j jWðtÞij2 ¼ jbj ðtÞj2 .
ð11Þ
The duration of the laser–molecule interaction, Dtint, is the time which a molecule from the molecular beam spends crossing the laser beam. So, in faster molecular beams or under shorter laser pulses the interaction time is shorter and the total interference picture differs from that for a longer duration. The phase control permits to manipulate with ion signal, Eq. (10), by means of a simple change of relative phase / of both laser waves. The modulation depth, which is equal to Fm ¼
imax imin ; imax þ imin
ð12Þ
where imax and imin are the maximum and minimum of the ion signal over relative laser phase, gives us a quantitative characteristic of the phase control. The detuned phase control is illustrated in this Letter for laser fields and molecular parameters corresponding to experiments [5,6] by means of direct numerical integration of the time-dependent matrix Eq. (6) (under cer-
! 2i½V 1 cosðx1 t þ /1 Þ þ V 3 cosðx3 t þ /3 Þ : ixj þ 12 cj
ð8Þ
tain approximations Eq. (6) can be integrated in a closed analytic form [15]). Fig. 2 presents the total ion signal, Eq. (10) calculated versus relative laser phase for different triple wave detunings. The evident phase control is displayed for the ideal and fine tripling where the detuning rate is k = 0, 107. An increase of tripling detuning destroys the control: the larger detuning shifts the phase of total ion signal, though the averaged value of ion signal increases. This shows that a mismatch of the third harmonic frequency from the triple fundamental one is a crucial parameter in phase control because it causes an additional time-dependent dephasing which, been averaged over time, damps the control 2. The control becomes worse already for the experimentally good tripling where k = 106 and the frequency mismatch dx = 3.13 ns1 becomes comparable with the unperturbed resonance detuning Dx, Dx = Dx2 = h1(j E0) 3x1 (dx is still much less than the Stark shift of the molecular levels induced by the fundamental laser). This result confirms the fact, that although the strong fundamental laser shifts the excited state, the resonance structure of unperturbed molecule is mainly responsible for efficiency of the phase control. Fig. 3 presents the ion signal and modulation depth, Eqs. (10) and (12), versus the fundamental laser wavelength calculated for different tripling detunings and excited state linewidths. Parameters of the model system were chosen in order to better fit the experimental situation [5,6] (the middle column of graphs) and its two limits: long-living excited state (more likely to experiments with diatomics and simple molecules) – the left column, and short-living excited state (complex polyatomics) – the right column. The numerical modulation depths on middle and right graphs show the behavior and values very like the experimental data [5,6]. The calculated absorption profiles near resonance are more narrow than the experimental ones due to the approximation taking the one excited state into account. An increase of the excited state linewidth supresses ion signal and modulation depth because a faster decay dumps excited state population and destroys the control. The modulation depth (in contrast to the ion signal) displays a drastic sensitivity to the triple wave mismatch mainly for wider excited states with c = 10, 100 ns1, where, even for a good tripling with
2
Within a perturbative approximation one simply derives this phase in an analytic form. This will be demonstrated in detail in a longer paper.
A.I. Pegarkov / Chemical Physics Letters 409 (2005) 8–13
ion signal, arb.units
9
k=0 -7 k=10 -6 k=10 -5 k=10 -4 k=10
8
7 0.03I(φ)
6
0
5
10
15
20
25
30
relative laser phase, rad Fig. 2. Total ion signal, Eq. (10) from two-level model molecule versus relative laser phase. The rate of triple wave detuning is: k = 0, 107, 106, 105, 104. The frequency difference between both molecular states is admitted as xj x0 = 3xres where kres = 601.8 nm. Intensities of the fundamental and third harmonics are I1 = 6.1 · 1010 W/cm2 and I3 = 3.9 · 103 W/cm2, respectively. sLaser = 5 ns = Dtint, Dx = 10 ns1, cj = 10 ns1.
k = 106, the modulation depth is 10 times less than that for ideal and fine triplings, k = 0, 107. For more stable excited state with c = 1 ns1 the phase control is less sensitive to tripling detuning: here all the calculated profiles show approximately the same wave length dependence. Such a behavior of the modulation depth arises due to competition of two different processes: a laser-induced excitation and a bath-induced decay. For a more -1
modulation depth
γ j =1 ns
γ j=10 ns
-1
γ j=100 ns
0.3
0.15
0.2
0.2
0.1
0.1
0.1
0.05
0
0
0
6
6
5
5
4
4
3
3
2
2
2
1
1
1
0 -7 10 -6 10
5 ion signal, nm
stable system, whose excited state lifetime is sj 1 ns, its interaction with the bath is weak and the decay to it is slow. Therefore, in the 5 ns laser pulse this excited state is kept undecayed for a time long enough comparably with pulse duration. The interaction with the bath does not considerably destroy the coherence between the laser radiation and the system. As the result, the phase control over stable molecular states is less sensitive to the decoherence inside controlling field and attains higher modulation depth. This effect is good confirmed by experiments [5,6] where ammonia exhibited better modulation depth with the maximum of 33% than trimethylamine, triethylamine, cyclooctaetraene, and 1,1-dimethylhydrazine which did only 22%. As it was earlier mentioned in [16] a change of the laser field intensity could spectacularly improve the phase control. Fig. 4 displays the modulation depth together with total ion signal for one-photon versus three-photon phase control in dependence on the fundamental laser intensity calculated, in contrast to [16], including the laser frequency mismatch. One can see that the change of the fundamental laser intensity between 1010 and 1012 W/cm2 considerably increases the modulation depth. The steep rise up of modulation depth occurs under the peak laser intensities where the bichromatic (x1, x3) absorption has a deep non-linear fall down. Within this intensity area the ions independently produced by the x1 and x3 monochromatic photoabsorptions differ from each other more than in 10 times. Fig. 4 presents another illustration of the role
0.3
6
4
-5
10 -4 10
3
0 601
602
11
0 0 603 601 602 603 wave length of fundamental laser, nm
601
-1
602
603
Fig. 3. Modulation depth, Eq. (12) and ion signal, Eq. (10) for two-level model molecule versus wavelength of fundamental laser for different linewidths of excited state and tripling detunings. The model parameters correspond to experiment [5,6]: laser pulse duration is 5 ns and others are as in Fig. 2. Ion signal is calculated for / = 0 rad.
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A.I. Pegarkov / Chemical Physics Letters 409 (2005) 8–13
(a) 0.4 k=0 -7 10 -6 10 -5 10 -4 10 δω=0, γj=0
modulation depth
0.3
0.2
0.1
0
(b) 2
ion signal, arb.units
10
0
10
ω1 ω3=3ω1 ω1 + ω3
-2
10
-4
10
8
10
12
14
10 10 10 10 2 peak intensity of fundamental laser, W/cm
Fig. 4. Modulation depth, Eq. (12) (a) and ion signal (b), Eq. (10) from two-level model molecule versus peak intensity of fundamental laser for different triple wave detunings, k = 0, 107, 106, 105, 104. The modulation depth without decay and with ideal tripling, k = 0, is shown as a dotted line in (a). Other parameters are as in Fig. 2.
playing by triple wave detuning: even for the good tripling with k = 106, dx 3.13 ns1 the maximum of modulation depth is six times less than for ideal tripling with k = 0 and almost suppressed for worse tripling.
4. Resume The quantum model of two levels with decay permits to analyze temporal dynamics of phase control over photoabsorption in molecules. The results show that in intense laser fields, where the perturbation theory is not longer valid, the bichromatic absorption probability cannot be counted like a simple interference from two individual monochromatic amplitudes but includes the higher-perturbative monochromatic amplitudes together with mixed bichromatic amplitudes. For an effective bichromatic control the frequency coherence of controlling laser waves is very important. The frequency mismatch between fundamental laser and its third harmonic creates an additional time-dependent phase which visibly damps the phase control. The molecular decay into a bath of excited molecular states
and continua also affects upon the control: the more stable molecules manifest a higher modulation depth. The control over less stable molecules is more sensitive to frequency decoherence in the controlling field than the control over more stable molecules: thus, a frequency mismatch in the controlling laser light causes less modulation depth in complex polyatomics than in simple molecules. From the above results one may conclude that for all the molecules studied in experiments [5,6] the tripling wave detuning reached by the apparatus was at least more than 2cj and probably around 10cj–30cj. The unusual modulation depth dependences shown in Figs. 5, 7–9 of paper [5] can be due to the triple wave detuning. The modulation depth for ammonia shown in Fig. 3 of paper [5] rather occurs due to a resonance with at least two different excited molecular states, for which the resonance laser detuning on the shorter laser waves lies beyond the molecular excited state linewidth. The intensity of fundamental laser is the key parameter of the phase control. Just changing the peak intensity between 1010 and 1012 W/cm2 one can drastically modify the modulation depth. In order to experimentally confirm the results obtained in this Letter new experiments like [2–6] can be performed with the same molecular compounds but (i) with different triple wave detuning coefficients and (ii) with shorter laser pulses when the pulse duration could be less than the excited state lifetime. The modulation depth measured in both new experiments should be different from the data presented in papers [5,6] and look like shown in this Letter.
Acknowledgments I am very thankful to Professor Paul W. Brumer for formulating me this problem and for many stimulating discussions. This work was supported by the Chemical Physics Theory Group at the University of Toronto.
References [1] M. Shapiro, J.W. Hepburn, P. Brumer, Chem. Phys. Lett. 149 (1988) 451. [2] S.P. Lu, S.M. Park, Y. Xie, R.J. Gordon, J. Chem. Phys. 96 (1992) 6613. [3] V.D. Kleiman, L. Zhu, X. Li, R.J. Gordon, J. Chem. Phys. 102 (1995) 5863. [4] L. Zhu, V. Kleiman, X. Li, S.P. Lu, K. Trentelman, R.J. Gordon, Science 207 (1995) 77. [5] X. Wang, R. Bersohn, K. Takahashi, M. Kawasaki, H.L. Kim, J. Chem. Phys. 105 (1996) 2992. [6] R. Bersohn, J. Mol. Struct. 480–481 (1999) 231.
A.I. Pegarkov / Chemical Physics Letters 409 (2005) 8–13 [7] E.D. Potter, J.L. Herek, S. Pedersen, Q. Liu, A.H. Zewail, Nature 355 (1992) 66. [8] R.J. Levis, G.M. Menkir, H. Rabitz, Science 292 (2001) 709. [9] T. Brixner, N.H. Damrauer, G. Gerber, Adv. At. Mol. Opt. Phys. 46 (2001) 287. [10] J. Herek, W. Wohlleben, R.J. Cogdell, D. Zeidler, M. Motzkus, Nature 417 (2002) 533. [11] D.S. Elliot, private communication.
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[12] H.M. van Driel, private communication. [13] M. Shapiro, P. Brumer, Principles of the Quantum Control of Molecular Processes, Wiley, Toronto, 2002. [14] B.W. ShoreTheory of Coherent Atomic Excitation, vol. 1, Wiley, New York, 1990. [15] H. Han, P. Brumer, Chem. Phys. Lett. 406 (2005) 237. [16] T. Nakajima, P. Lambropoulos, S. Cavalieri, M. Matera, Phys. Rev. A 46 (1992) 7315.
Phase control effects induced by a detuning of third laser harmonic from triple fundamental frequency Alexander I. Pegarkov
*,1
Chemical Physics Theory Group, Department of Chemistry, University of Toronto, 80 St. George Street, Toronto, Ont., Canada M5S 3H6 Received 7 February 2005; in final form 5 April 2005 Available online 23 May 2005
Abstract The phase control of the absorption of three photons from the fundamental laser mode and one photon from its third harmonic is studied allowing for a small difference between frequency of the third harmonic and the triple frequency of the fundamental wave. The effect of triple wave detuning upon the (x1, x3) absorption is illustrated numerically for model molecule NH3. It is shown that in experiments the phase control can be affected by such a frequency detuning. Several experimental schemes are discussed in order to check out and to confirm this effect. 2005 Elsevier B.V. All rights reserved.
1. Introduction More than 15 years ago, the first theoretical study [1] showed that the bichromatic excitation provided by the fundamental laser line and its third harmonic could afford an experimentally simple route to control over unimolecular reactions. Since that time the experimental methods have been excellently developed and permit now to control the (x1, 3x1) excitation and ionization of simple and large polyatomic molecules by a change of either relative phase or intensity of fundamental and third laser harmonics [2–6] . In last decade due to a fast development in the technique of short-pulse lasers the strong experimental and theoretical efforts have been making also to use the femtosecond pulses of powerful lasers in order to
*
Fax: +1 613 562 5190. E-mail address: [email protected]. 1 Present address (since January 1, 2004): Department of Physics, University of Ottawa, 150 Louis Pasteur, Ottawa, Ont., Canada K1N 6N5. 0009-2614/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2005.04.089
achieve effective control over photochemical processes [7–10]. In contrast to the laser beam experiments [2–6], a linear dispersion of the frequency-multiplying medium, and a frequency chirp create also some mismatch between frequency of the nÕs harmonic and n multiplied the fundamental laser frequency, which can change the output of a chemical process controlled by pulsed laser. In experiments using pulsed lasers one gets the third harmonic of the frequency slightly different from the triple fundamental frequency [11,12]: x3 = 3x1 + dx, Fig. 1. Actually the detuning of a triple wave dx is very small like (106 104)x1 and was never taken into account in theoretical and experimental analysis. The goal of this Letter is to treat the one-photon versus three-photon control scheme in the case of triple wave detuning: (x1, 3x1 + dx). The results demonstrate that, although the frequency detuning is small, it visibly impacts upon the phase control and changes the chemical product output. The inclusion of triple wave detuning into the quantum control theory [13] can help to explain an unusual modulation depth behavior observed earlier in experiments [5,6].
A.I. Pegarkov / Chemical Physics Letters 409 (2005) 8–13
9
where H0 is the unperturbed molecular Hamiltonian, HD is a phenomenological anti-hermitian operator causing a laser-free decay to the bath, V1(t) and V3(t) are the operators of dipole interaction of the molecule with the fundamental laser wave and its third harmonic V 1;3 ðtÞ ¼ ðd; E1;3 Þf ðs; tÞ cosðx1;3 t þ /1;3 Þ; d is the molecular dipole moment and f(s, t) is the pulse shape function, 1; 0 6 t 6 s; f ðs; tÞ ¼ 0; t < 0; t > s.
Fig. 1. Excitation diagram for phase control with two interfering amplitudes created by three-photon and one-photon transitions from the ground molecular state. (a) The ideal case, where the frequency of the third harmonic, x3 is sharply equal to the triple frequency of the fundamental laser line, x3 = 3x1. (b) A realistic case, where x3 and 3x1 are slightly detuned, x3 = 3x1 + dx.
Let us consider a two-level quantum system irradiated by a laser wave x1 and its third harmonic x3 having a frequency detuning (Fig. 1b) EðtÞ ¼ E1 cosðx1 t þ /1 Þ þ E3 cosðx3 t þ /3 Þ; dx ¼ kx1
and assume that the initial molecular state is the ground electronic-rotation–vibration state while the final state is an excited one having an energy width due to various non-radiative transitions to a bath of other excited molecular states and continua Ej ¼ j iCj =2. In polyatomic molecules these transitions are much faster than spontaneous decay to the ground state, which is not included here. The quantum system, been excited to state |jæ, can then absorb two photons from fundamental mode and reach ionization continuum. In experiment the dynamics of resonance enhanced multiphoton ionization repeats the population dynamics of excited molecular state. This model therefore does not include ionization continuum and assumes that the direct laser-induced transitions from both states to the bath are forbidden as well. The temporal dynamics of the one photon versus three photon phase control is described by the following time-dependent Schro¨dinger equation oWðtÞ ¼ H ðtÞWðtÞ; ot H ðtÞ ¼ H 0 þ H D þ V 1 ðtÞ þ V 3 ðtÞ;
ih
U0 ðtÞ ¼ eix0 t j/0 i; Uj ðtÞ ¼ eiðxj icj =2Þt j/j i;
ð2Þ
x0 ¼ E0 =h; xj ¼ j =h; cj ¼ Cj =h; where the functions |/0æ, |/jæ fit the following time-independent Schro¨dinger equations
2. Coherent control theory for the detuned (x1, x3) absorption to a decaying quantum state
x3 ¼ 3x1 þ dx;
The laser-free states of our two-level model molecule are the quasi-stationary states which obey the wave equation like Eq. (1) having a time-independent laserfree Hamiltonian. The laser-free quasi-stationary states can be written as
ð1Þ
½H 0 þ H D j/0 i ¼ E0 j/0 i; Cj ½H 0 þ H D j/j i ¼ j i j/j i. 2
ð3Þ ð4Þ
Before the laser pulse reaches the model molecule, the molecule is in its ground state. As the laser field begins to interact with the molecule, then both molecular states are populated and the population maximum oscillates between both molecular states. Once two laser modes have different frequencies, the population oscillation frequency differs from the Rabi frequencies independently induced by each laser mode and the full population dynamics displays a complex pattern created by two time-dependent perturbations. At a time t during the laser–molecule interaction, 0 6 t 6 Dtint, the total time-dependent state can be represented by the following superposition WðtÞ ¼ b0 ðtÞj/0 i þ bj ðtÞj/j i.
ð5Þ
Substituting Eq. (5) into Eq. (1), multiplying it by the proper bra- state vectors and having in mind Eqs. (3) and (4), we obtain the following matrix equation for the time-dependent state amplitudes b0(t) and bj(t) d BðtÞ ¼ HðtÞBðtÞ; dt B(t) is the state amplitude vector b0 ðtÞ BðtÞ ¼ bj ðtÞ
ð6Þ
ð7Þ
and H(t) is the following non-diagonal symmetric matrix
10
A.I. Pegarkov / Chemical Physics Letters 409 (2005) 8–13
HðtÞ ¼
ix0 2i½V 1 cosðx1 t þ /1 Þ þ V 3 cosðx3 t þ /3 Þ
The initial condition for Eq. (6) is 1 Bðt ¼ 0Þ ¼ . 0
ð9Þ
Finally, Eqs. (6)–(9) present a phase control version of the time-dependent Schro¨dinger equation for a two-level system with decay [14].
3. Phase modulation of the detuned (x1, x3) absorption: numerical results and discussion Let us show that the time-dependent quantum interference of the slightly incoherent x3 and 3x1 pathways significantly impacts upon the phase control and can be a possible source of unusual phase modulation observed experimentally in [5,6]. In [5,6] the ion signal of two-photon ionization from the excited |jæ state has been measured. The ions are produced in the point of intersection of the laser beam and molecular beam. Therefore, the ion signal generated by this way is proportional to the |j æ |0æ bichromatic absorption probability Pj(/, t) integrated over the time of the laser–molecule interaction Z Dtint ið/Þ P j ð/; tÞ dt; ð10Þ 0
P j ð/; tÞ jh/j jWðtÞij2 ¼ jbj ðtÞj2 .
ð11Þ
The duration of the laser–molecule interaction, Dtint, is the time which a molecule from the molecular beam spends crossing the laser beam. So, in faster molecular beams or under shorter laser pulses the interaction time is shorter and the total interference picture differs from that for a longer duration. The phase control permits to manipulate with ion signal, Eq. (10), by means of a simple change of relative phase / of both laser waves. The modulation depth, which is equal to Fm ¼
imax imin ; imax þ imin
ð12Þ
where imax and imin are the maximum and minimum of the ion signal over relative laser phase, gives us a quantitative characteristic of the phase control. The detuned phase control is illustrated in this Letter for laser fields and molecular parameters corresponding to experiments [5,6] by means of direct numerical integration of the time-dependent matrix Eq. (6) (under cer-
! 2i½V 1 cosðx1 t þ /1 Þ þ V 3 cosðx3 t þ /3 Þ : ixj þ 12 cj
ð8Þ
tain approximations Eq. (6) can be integrated in a closed analytic form [15]). Fig. 2 presents the total ion signal, Eq. (10) calculated versus relative laser phase for different triple wave detunings. The evident phase control is displayed for the ideal and fine tripling where the detuning rate is k = 0, 107. An increase of tripling detuning destroys the control: the larger detuning shifts the phase of total ion signal, though the averaged value of ion signal increases. This shows that a mismatch of the third harmonic frequency from the triple fundamental one is a crucial parameter in phase control because it causes an additional time-dependent dephasing which, been averaged over time, damps the control 2. The control becomes worse already for the experimentally good tripling where k = 106 and the frequency mismatch dx = 3.13 ns1 becomes comparable with the unperturbed resonance detuning Dx, Dx = Dx2 = h1(j E0) 3x1 (dx is still much less than the Stark shift of the molecular levels induced by the fundamental laser). This result confirms the fact, that although the strong fundamental laser shifts the excited state, the resonance structure of unperturbed molecule is mainly responsible for efficiency of the phase control. Fig. 3 presents the ion signal and modulation depth, Eqs. (10) and (12), versus the fundamental laser wavelength calculated for different tripling detunings and excited state linewidths. Parameters of the model system were chosen in order to better fit the experimental situation [5,6] (the middle column of graphs) and its two limits: long-living excited state (more likely to experiments with diatomics and simple molecules) – the left column, and short-living excited state (complex polyatomics) – the right column. The numerical modulation depths on middle and right graphs show the behavior and values very like the experimental data [5,6]. The calculated absorption profiles near resonance are more narrow than the experimental ones due to the approximation taking the one excited state into account. An increase of the excited state linewidth supresses ion signal and modulation depth because a faster decay dumps excited state population and destroys the control. The modulation depth (in contrast to the ion signal) displays a drastic sensitivity to the triple wave mismatch mainly for wider excited states with c = 10, 100 ns1, where, even for a good tripling with
2
Within a perturbative approximation one simply derives this phase in an analytic form. This will be demonstrated in detail in a longer paper.
A.I. Pegarkov / Chemical Physics Letters 409 (2005) 8–13
ion signal, arb.units
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k=0 -7 k=10 -6 k=10 -5 k=10 -4 k=10
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7 0.03I(φ)
6
0
5
10
15
20
25
30
relative laser phase, rad Fig. 2. Total ion signal, Eq. (10) from two-level model molecule versus relative laser phase. The rate of triple wave detuning is: k = 0, 107, 106, 105, 104. The frequency difference between both molecular states is admitted as xj x0 = 3xres where kres = 601.8 nm. Intensities of the fundamental and third harmonics are I1 = 6.1 · 1010 W/cm2 and I3 = 3.9 · 103 W/cm2, respectively. sLaser = 5 ns = Dtint, Dx = 10 ns1, cj = 10 ns1.
k = 106, the modulation depth is 10 times less than that for ideal and fine triplings, k = 0, 107. For more stable excited state with c = 1 ns1 the phase control is less sensitive to tripling detuning: here all the calculated profiles show approximately the same wave length dependence. Such a behavior of the modulation depth arises due to competition of two different processes: a laser-induced excitation and a bath-induced decay. For a more -1
modulation depth
γ j =1 ns
γ j=10 ns
-1
γ j=100 ns
0.3
0.15
0.2
0.2
0.1
0.1
0.1
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0
0
0
6
6
5
5
4
4
3
3
2
2
2
1
1
1
0 -7 10 -6 10
5 ion signal, nm
stable system, whose excited state lifetime is sj 1 ns, its interaction with the bath is weak and the decay to it is slow. Therefore, in the 5 ns laser pulse this excited state is kept undecayed for a time long enough comparably with pulse duration. The interaction with the bath does not considerably destroy the coherence between the laser radiation and the system. As the result, the phase control over stable molecular states is less sensitive to the decoherence inside controlling field and attains higher modulation depth. This effect is good confirmed by experiments [5,6] where ammonia exhibited better modulation depth with the maximum of 33% than trimethylamine, triethylamine, cyclooctaetraene, and 1,1-dimethylhydrazine which did only 22%. As it was earlier mentioned in [16] a change of the laser field intensity could spectacularly improve the phase control. Fig. 4 displays the modulation depth together with total ion signal for one-photon versus three-photon phase control in dependence on the fundamental laser intensity calculated, in contrast to [16], including the laser frequency mismatch. One can see that the change of the fundamental laser intensity between 1010 and 1012 W/cm2 considerably increases the modulation depth. The steep rise up of modulation depth occurs under the peak laser intensities where the bichromatic (x1, x3) absorption has a deep non-linear fall down. Within this intensity area the ions independently produced by the x1 and x3 monochromatic photoabsorptions differ from each other more than in 10 times. Fig. 4 presents another illustration of the role
0.3
6
4
-5
10 -4 10
3
0 601
602
11
0 0 603 601 602 603 wave length of fundamental laser, nm
601
-1
602
603
Fig. 3. Modulation depth, Eq. (12) and ion signal, Eq. (10) for two-level model molecule versus wavelength of fundamental laser for different linewidths of excited state and tripling detunings. The model parameters correspond to experiment [5,6]: laser pulse duration is 5 ns and others are as in Fig. 2. Ion signal is calculated for / = 0 rad.
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A.I. Pegarkov / Chemical Physics Letters 409 (2005) 8–13
(a) 0.4 k=0 -7 10 -6 10 -5 10 -4 10 δω=0, γj=0
modulation depth
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(b) 2
ion signal, arb.units
10
0
10
ω1 ω3=3ω1 ω1 + ω3
-2
10
-4
10
8
10
12
14
10 10 10 10 2 peak intensity of fundamental laser, W/cm
Fig. 4. Modulation depth, Eq. (12) (a) and ion signal (b), Eq. (10) from two-level model molecule versus peak intensity of fundamental laser for different triple wave detunings, k = 0, 107, 106, 105, 104. The modulation depth without decay and with ideal tripling, k = 0, is shown as a dotted line in (a). Other parameters are as in Fig. 2.
playing by triple wave detuning: even for the good tripling with k = 106, dx 3.13 ns1 the maximum of modulation depth is six times less than for ideal tripling with k = 0 and almost suppressed for worse tripling.
4. Resume The quantum model of two levels with decay permits to analyze temporal dynamics of phase control over photoabsorption in molecules. The results show that in intense laser fields, where the perturbation theory is not longer valid, the bichromatic absorption probability cannot be counted like a simple interference from two individual monochromatic amplitudes but includes the higher-perturbative monochromatic amplitudes together with mixed bichromatic amplitudes. For an effective bichromatic control the frequency coherence of controlling laser waves is very important. The frequency mismatch between fundamental laser and its third harmonic creates an additional time-dependent phase which visibly damps the phase control. The molecular decay into a bath of excited molecular states
and continua also affects upon the control: the more stable molecules manifest a higher modulation depth. The control over less stable molecules is more sensitive to frequency decoherence in the controlling field than the control over more stable molecules: thus, a frequency mismatch in the controlling laser light causes less modulation depth in complex polyatomics than in simple molecules. From the above results one may conclude that for all the molecules studied in experiments [5,6] the tripling wave detuning reached by the apparatus was at least more than 2cj and probably around 10cj–30cj. The unusual modulation depth dependences shown in Figs. 5, 7–9 of paper [5] can be due to the triple wave detuning. The modulation depth for ammonia shown in Fig. 3 of paper [5] rather occurs due to a resonance with at least two different excited molecular states, for which the resonance laser detuning on the shorter laser waves lies beyond the molecular excited state linewidth. The intensity of fundamental laser is the key parameter of the phase control. Just changing the peak intensity between 1010 and 1012 W/cm2 one can drastically modify the modulation depth. In order to experimentally confirm the results obtained in this Letter new experiments like [2–6] can be performed with the same molecular compounds but (i) with different triple wave detuning coefficients and (ii) with shorter laser pulses when the pulse duration could be less than the excited state lifetime. The modulation depth measured in both new experiments should be different from the data presented in papers [5,6] and look like shown in this Letter.
Acknowledgments I am very thankful to Professor Paul W. Brumer for formulating me this problem and for many stimulating discussions. This work was supported by the Chemical Physics Theory Group at the University of Toronto.
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A.I. Pegarkov / Chemical Physics Letters 409 (2005) 8–13 [7] E.D. Potter, J.L. Herek, S. Pedersen, Q. Liu, A.H. Zewail, Nature 355 (1992) 66. [8] R.J. Levis, G.M. Menkir, H. Rabitz, Science 292 (2001) 709. [9] T. Brixner, N.H. Damrauer, G. Gerber, Adv. At. Mol. Opt. Phys. 46 (2001) 287. [10] J. Herek, W. Wohlleben, R.J. Cogdell, D. Zeidler, M. Motzkus, Nature 417 (2002) 533. [11] D.S. Elliot, private communication.
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[12] H.M. van Driel, private communication. [13] M. Shapiro, P. Brumer, Principles of the Quantum Control of Molecular Processes, Wiley, Toronto, 2002. [14] B.W. ShoreTheory of Coherent Atomic Excitation, vol. 1, Wiley, New York, 1990. [15] H. Han, P. Brumer, Chem. Phys. Lett. 406 (2005) 237. [16] T. Nakajima, P. Lambropoulos, S. Cavalieri, M. Matera, Phys. Rev. A 46 (1992) 7315.