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New quantum control pathway for a coupled-potential system
Chemical Physics Letters 369 (2003) 525–533 www.elsevier.com/locate/cplett
New quantum control pathway for a coupled-potential system Yukiyoshi Ohtsuki *, Kazuki Ohara, Mayumi Abe, Kazuyuki Nakagami 1, Yuichi Fujimura Department of Chemistry, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan Received 16 August 2002; in final form 26 October 2002
Abstract Optimal control theory is used to design laser pulses that enhance the trans–cis photoisomerization of retinal, which is modeled by coupled potential curves to examine coupling effects on control mechanisms. Based on numerical results, a new type of pump–dump control pathway, in which an optimal pulse achieves high target probability without directly modifying a potential coupling, is proposed. The key feature of this control is that a shaped wave packet passes a curvecrossing point adiabatically in the ground electronic state. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction Non-adiabatic coupling between potential energy surfaces plays an essential role in various photochemical processes, including dissociation, isomerization, electron transfer, and other photochemical reactions [1–3]. It is thus expected that the control of non-adiabatic transitions by means of shaped laser excitation would enable selective induction of subsequent photochemical events with high efficiency. According to this idea, the possibility of controlling branching ratios has been *
Corresponding author. E-mail address: [email protected] (Y. Ohtsuki). 1 Present address: Nippon Timeshare Co. LTD., 3-11-24 Minato-ku, Tokyo 108-8561, Japan.
discussed on the basis of optimal control theory using curve-crossing systems that typically model diatomic molecules [4–10]. Their control pathways were shown to undergo considerable changes depending on the potential shape as well as the control objective. In this Letter, we focus on the trans–cis isomerization of the retinal chromophore in bacteriorhodopsin, which has different potential curves from those studied so far [11]. We show that this isomerization can be controlled by a new type of pump–dump mechanism that utilizes the adiabatic motion of a wave packet in the ground electronic state. It is thought that the photoisomerization of retinal can be described by coupled low-dimensional potentials [11–15], although a thorough understanding of the reaction mechanism has not been attained. This deduction is justified by partial
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(02)02030-4
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dynamic guidance provided by surrounding media, which achieves efficient isomerization. In fact, the potential surfaces relevant to photoisomerization are often modeled by two- or even one-dimensional potentials [11–15], the parameters of which are determined to reproduce experimentally observed spectra. For the sake of comparison, we briefly summarize herein the results of previous theoretical studies in which optimal control pathways for potential crossings of small molecules were examined. Gross et al. [4,5] studied a system with three electronic states, two of which were dissociative electronic excited states coupled with each other. Optimal pulses were designed in order for an initially prepared wave packet to remain or change the original potential after passing a crossing region. They showed that in a diabatic-transition case, an optimal pulse utilizes the optical transitions between the ground and excited states to increase the average energy of a wave packet (the Landau–Zener mechanism). To induce adiabatic transitions, on the other hand, an optimal pulse transfers a wave packet into a third state, and the packet is pumped back to one of the excited states after passing the crossing region. On the other hand, we studied predissociation of NaI using a two-orientation model with the aim of accelerating the predissociation [8,9]. We found two optimal control pathways for this target, i.e., a pump– dump mechanism and the Landau–Zener mechanism, in which excited wave packets are controlled to move along the potential curve adiabatically and diabatically, respectively. Recently, Vivie-RiedelÕs group [10] dealt with the optimal control of a system with a conical intersection and designed a pulse that creates a localized wave packet around the conical intersection. This Letter is organized as follows. In Section 2, we introduce our model potential and coupled pulse design equations. The numerical results and a summary are given in Section 3.
2. Theory We adopt a one-dimensional model for the excited state trans–cis isomerization of a retinal
chromophore in bacteriorhodopsin. This simplified model was proposed by Pollard et al. [11] in order to qualitatively reproduce transient absorption spectra. Here we slightly modify their potential [11] to discuss the effects of non-adiabatic transitions on the laser control of the isomerization. As shown in Fig. 1, we assume two electronic states whose diabatic potential curves, V11 ðhÞ and V22 ðhÞ, are modeled by 50 cm1 harmonic oscillators with minima at dihedral angles of h ¼ 0° and h ¼ 180° corresponding to trans- and cis-isomers, respectively. These two diabatic states interact with each other around the potential crossing whose interaction potential, V12 ðhÞ ½V21 ðhÞ, is approximated by a Gaussian function centered at a crossing point and having a width of Dh ¼ 10°. This molecular system interacts with a timedependent electric field, EðtÞ, through the semiclassical dipole–field interaction. Since it is known that the coherent motion of the excited wave packet is observed during the isomerization process [16], the time evolution of the system is assumed to be described by the Schr€ odinger equation. Within our qualitative treatment of isomerization, this assumption may hold as long as we are concerned with ultrafast (subpicosecond) dynamics. Then, our system obeys the equation of motion:
Fig. 1. Model potential energy curves in an adiabatic representation. A new pump- and dump-control pathway is schematically illustrated by the wave packets and arrows. The solid and dotted arrows indicate optical transitions and wave packet propagation on the ground-state potential, respectively.
Y. Ohtsuki et al. / Chemical Physics Letters 369 (2003) 525–533
" # o w1 ðh; tÞ ih ot w2 ðh; tÞ " # V12 ðhÞ l12 ðhÞEðtÞ T þ V11 ðhÞ ¼ V21 ðhÞ l21 ðhÞEðtÞ T þ V22 ðhÞ " # w1 ðh; tÞ ; ð1Þ w2 ðh; tÞ where wk ðh; tÞ is the wavefunction of the kth diabatic state (k ¼ 1; 2), and l21 ðhÞ [l12 ðhÞ] is the matrix element of the dipole moment operator, lðhÞ, which is associated with the w1 ! w2 [w2 ! w1 ] transition. They are expressed as l21 ðhÞ ¼ l12 ðhÞ ¼ l0 ðhÞ cos h for convenience. Here we assume that the electric field is linearly polarized. An optimal control pulse is designed according to a standard procedure [17]. A target operator, W, is introduced to specify a physical objective. An optimal pulse is designed so that it maximizes an expectation value of a target operator at a final time, tf , subject to minimum pulse fluence. Then, the optimal pulse is defined by a pulse that gives a maximal value to the objective functional: Z tf 1 2 J ¼ hwðtf ÞjW jwðtf Þi dt½EðtÞ hA 0
Z tf o i t ; 2Re dt nðtÞ þ H wðtÞ ot h 0 ð2Þ wherejwðtÞi and H t denote the total wavefunction and the total Hamiltonian, respectively, the positive constant A, which is called the weight parameter, weighs the significance of the penalty, and jnðtÞi is the total Lagrange multiplier constraining the system to obey the equation of motion, Eq. (1). Applying calculus of variations to Eq. (2), we obtain the following coupled pulse design equations. The optimal pulse is expressed as EðtÞ ¼ A ImfhnðtÞjlðhÞjwðtÞig:
ð3Þ
The time evolution of the total Lagrange multiplier is determined by integrating the equation of motion o i h jnðtÞi ¼ H t jnðtÞi; ð4Þ ot
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with the final condition of jnðtf Þi ¼ W jwðtf Þi:
ð5Þ
Here, the Hamiltonian in Eq. (4), H t , is the total Hamiltonian whose expression is given in Eq. (1).
3. Results and discussion In order to examine the effects of non-adiabatic interaction on an optimal pathway for photoisomerization, we consider two cases, i.e., strong- and medium-coupling cases. We first illustrate these coupling strengths by calculating the time-dependent behavior of a Franck–Condon packet in the absence of electric fields. The Franck–Condon packet is a packet that has the same form as that of the initial state but is on an excited surface, V22 ðhÞ. Throughout this Letter, we assume that the molecular system is initially in the lowest vibrational state of the all-trans isomer. In the strong-coupling (adiabatic) case, 96.5% of the initial population of the Franck–Condon packet adiabatically switches to another diabatic potential after initially passing the crossing point. In the medium-coupling case, on the other hand, 44.5% of the initial population remains in the original diabatic potential, V22 ðhÞ, due to the non-adiabatic interaction. Our purpose here is to design an optimal pulse that transfers as much of the initial state (the lowest vibrational state of the all-trans isomer) as possible to the cis-isomer. For this purpose, we adopt a target operator that projects the subspace spanned by the lowest six vibrational states associated with a cis-isomer, {jcis vi}, W ¼
5 X
jcis vihcis vj;
ð6Þ
v¼0
where the energy eigenvalues of these states are far below the potential barrier that separates the two isomers. Optimal pulses are obtained by solving the pulse design equations together with this target operator and the amplitude parameter, A ¼ 4:0 106 . In simulations shown below, the final time is set to tf ¼ 500 fs, which is shorter than the period of the wave packet oscillation in the excited adiabatic state. This time interval is divided into
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Nt ¼ 100,000 time steps to numerically integrate the equations of motion, Eqs. (1) and (4). We prevent a wave packet from rotating beyond 2p by removing a periodic boundary condition from the potential curves [11]. As a tradeoff, we assume an artificial boundary condition in which a fictitious angle range 3p is used to avoid wave packet reflection at the edge of the grid. Then, adopting 1024 uniformly spaced grid basis, the time evolution of the wave packet is calculated by the secondorder, split-operator method with the fast Fourier transform (FFT) technique [18]. All of the potential couplings are calculated using the Pauli matrix [4]. In the first two numerical examples (Figs. 2 and 3), we assume a constant value of l0 ðhÞ ¼ l0 . For an iteration algorithm, we combine the method proposed by Zhu and Rabitz [19] and that proposed by Tannor et al. [20]. Since the algorithm proposed by Zhu and Rabitz [19] has high nu-
Fig. 3. Results in the medium-coupling case. (a) The optimal control pulse as a function of time and (b) the time evolution of the target probability (solid line) and that of the populations on the ground (dotted line) and excited (dot-dashed line) adiabatic states. The dashed line represents the time evolution of the target probability calculated using the filtered pulse.
Fig. 2. Results in the strong-coupling (adiabatic) case. (a) The optimal control pulse as a function of time and (b) the time evolution of the target probability (solid line) and that of the populations on the ground (dotted line) and excited (dotdashed line) adiabatic states.
merical reliability but sometimes suffers from slow convergence, we use this algorithm for the first 1000 iteration steps, by which the shape of an initial input pulse is adjusted so that it is close to an optimal solution. We then use the Krotov method [20] to save CPU time, since its rough search improves the speed of convergence near the optimal solution. In our calculations, a pulse is regarded as an optimal solution when the difference between the values of J in the final two steps is DJ =J 6 3:9 106 . Fig. 2 shows (a) the optimal pulse and (b) the temporal behavior of the target probability and that of the population on each adiabatic potential in the strong-coupling (adiabatic) case. This optimal pulse transfers 97.2% of the population to the target state [Eq. (6)]. The pulse consists of pump, shaping and dump subpulses, although we cannot
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clearly distinguish them due to the rather temporally broad and overlapped structure of the control pulse. The modulated subpulse in the time interval t 2 ½0 fs; 170 fs transfers the initial population into the electronic excited state, simultaneously adjusting the shape of the excited wave packet. We see from Fig. 2b that the excited packet then moves along the adiabatic potential during t 2 ½170 fs; 290 fs. Note that the total excited(ground-) state population does not change during this time interval, although we see intense pulse sequences (t 230 fs) in Fig. 2a. This means that population exchange processes between the two electronic states must occur. Through these processes, the shape of the wave packet is gradually adjusted, which is appropriate for the following dumping process. Such a wave packet shaping mechanism through population exchange processes was numerically observed in a different system with a different physical objective in our previous study [21]. We thus conclude that wave packet shaping through population exchange processes is essential for achieving a high control yield in a multi-electronic system. Finally, a subpulse during t 2 ½290 fs; 370 fs transfers the wave packet into the target state. The large temporal width of this dump pulse indicates that the wave packet has a low velocity and a spatially broadened structure around the Franck–Condon region accessible from the target state. Next, we consider the medium-coupling case. The calculated pulse and the time evolution of the target probability and that of the population on each adiabatic state are shown in Figs. 3a,b, respectively. This optimal pulse is composed of three well-separated subpulses and transfers 95.6% of the population to the target state. The first subpulse not only transfers the initial population to the electronic excited state but also adjusts the shape of the wave packet. Immediately after the excitation, the excited packet has nearly zero velocity and requires some time to leave the Franck– Condon region, thus providing sufficient time for the pulse to shape the packet. The combination of pump and shaping pulses results in a modulation structure in the first subpulse. The second subpulse may be called a half-cycle pulse. The electric field with a positive amplitude strengthens the diabatic
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potential coupling around the potential crossing (h ¼ 110°) and prevents the excited packet from moving along the diabatic potential V22 ðhÞ. By these first two subpulses, a localized wave packet is created on the diabatic potential V11 ðhÞ around the Franck–Condon region associated with the target state. The third subpulse efficiently transfers this localized packet into the target state specified by the operator in Eq. (6). Since the packet still has finite velocity at this time, the subpulse has to complete the dumping process very quickly, which is the reason why the dump pulse has a sharp temporal width. The feature of the optimal pulse in Fig. 3a is that it includes a low-frequency subpulse (a halfcycle pulse). To determine the role of this subpulse, we calculate the time evolution of the wave packet using a modified pulse that is obtained by eliminating low-frequency components from the original pulse by Fourier filtering. The time-dependent behavior of the expectation value of the target operator is shown by a dashed line in Fig. 3b, in which the target population at the final time is reduced to 52.0%. This means that it is essential to utilize a low-frequency pulse in order to obtain high target probability, as long as we adopt a conventional pump–dump control scheme. However, such a half-cycle pulse with high intensity and definite phase is not experimentally feasible at present, although half-cycle pulses with much lower intensities than that in Fig. 3a can be generated by the several pulse compression techniques relevant to their temporal widths [22,23]. The question is whether there is an alternative control pathway that achieves high target probability without using low-frequency pulses. To answer this question, we design an optimal pulse that does not include low-frequency components. To filter out the low-frequency components from the optimal pulse, Gross et al., for example, proposed a gradient filtering procedure. Here we will adopt an alternative convenient procedure [9] that is valid as long as the so-called frozen wave packet assumption [24,25] is fulfilled during the optical interaction. To outline our method, consider the case where a molecular system is excited by a pulse EðtÞ that has a small temporal width, Dt, and is centered at t ¼ 0. Note that under the frozen wave
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packet assumption [24,25], the kinetic energy operator is ignored in the propagator associated with the molecular Hamiltonian, resulting in the following approximate expression: expðiH Dt= hÞ ’ expðiV Dt= hÞ;
ð7Þ
where Dt is the period of the optical interaction and V is the total molecular potential. Utilizing Eq. (7) and the rotating wave approximation (RWA), the formal solution to Eq. (1), jwI ðtÞi, is expressed in the interaction picture as jwI ðtÞi jw0 i Z 1 h ’ 2Dt dx l21 ðhÞE~ðxÞ expfi½U21 ðhÞ xDtg 1 i þ h:c: jwI ðDtÞi; ð8Þ where jw0 i is the initial state, hU21 ðhÞ ¼ V22 ðhÞ V11 ðhÞ, and E~ðxÞ is a Fourier component of the pulse. When the pulse does not induce rapid Rabi oscillations, the wavefunction in Eq. (8) is a slowly varying function of time (the interaction picture). Thus, only the frequency components that satisfy the relation jV22 ðhÞ V11 ðhÞ hxj 6 h=Dt
ð9Þ
can induce optical transitions; that is, we can restrict frequency components in the optimal pulse by imposing restrictions on the optical transition region through the transition moment function, l0 ðhÞ [9]. According to this filtering procedure, we tentatively replace l0 ðhÞ ¼ l0 with 1 l0 ðhÞ ¼ l0 1 þ exp½aðh h1 Þ
1 þ ; ð10Þ 1 þ exp½aðh h2 Þ where a ¼ 0:5, h1 ¼ hcross 30:0, and h2 ¼ hcross þ 30:0 with the crossing point hcross ¼ 110:0°. Using Eq. (10), we solve the pulse design equations. Note that because l0 ðhÞ ¼ 0 around the potential crossing, this optimal pulse cannot directly modify the potential coupling. Fig. 4 shows (a) the calculated pulse, and (b) the time evolution of the target state and that of the population on each adiabatic potential. In these calculations, we found
Fig. 4. Results in the case where a pulse is designed with restricted frequency components. (a) The optimal control pulse as a function of time and (b) the time evolution of the target probability (solid line) and that of the populations on the ground (dotted line) and excited (dot-dashed line) adiabatic states.
that the pulse shown in Fig. 4a could realize virtually the same control scheme without assuming the artificial transition moment function given by Eq. (10). This was actually confirmed by numerical calculations with use of the original constant transition moment, l0 ðhÞ ¼ l0 . This means that our filtering procedure based on the frozen wave packet assumption offers a convenient alternative to eliminate low-frequency components from the optimal pulse. From Fig. 4a, we can see that the optimal pulse achieves a high isomerization probability of 90.0% without directly controlling the potential coupling but by adopting a considerably different control pathway from that in Fig. 3. The difference in control pathways can be seen by comparing the time-dependent behavior of the populations in Fig. 3b with that in Fig. 4b. In Fig. 3b, the wave
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packet moves on the excited adiabatic potential during the time interval, t 2 ½110 fs; 280 fs, whereas, in Fig. 4b, the packet moves on the ground potential. Based on these features, along with time- and frequency-resolved spectra (not shown here), we deduced the following new control mechanism. Before t 70 fs, the pulse creates a shaped wave packet on the excited adiabatic potential through population exchange processes. This excited packet is then transferred to high vibrational states in the ground state at t 100 fs. The average energy of this packet is close to the
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top of the potential barrier that separates the two isomers. Therefore, when passing the crossing region, the packet tends to move along the adiabatic potential because of its low velocity. After passing the crossing point, the packet continues to move toward the potential well associated with the cisisomer. Immediately before the packet reaches the target region, the pulse around t 300 fs pumps the wave packet up to the excited electronic state again. This packet is immediately transferred to the target state in the ground electronic state around t 350 fs.
Fig. 5. Time evolution of the wave packets in the excited (upper figure) and ground (lower figure) adiabatic states (a) in the case of Fig. 3 and (b) in the case of Fig. 4.
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To more clearly show the new control pathway, the time evolution of the wave packet on each adiabatic potential is presented in Fig. 5. For the sake of comparison, we show time evolutions of wave packets (a) in the case of Fig. 3 and (b) in the case of Fig. 4. In Fig. 5a, we see a typical wave packet motion induced by a pair of conventional pump and dump control pulses, although a modulated structure can also be seen around t 2 ½120 fs; 150 fs corresponding to packet-shaping processes. On the other hand, in the lower figure of Fig. 5b, the shaped wave packet appears around h 50° at t 70 fs in the ground (adiabatic) electronic state. The velocity of this packet gradually decreases as the packet climbs up the potential hill, and after passing the curve-crossing point, the velocity increases again. In the upper figure, we see that a very small portion of the wave packet is transferred to the excited state due to non-adiabatic coupling, which appears and remains around h 110° in the excited state. At the final stage around t ½250 fs; 350 fs, we observe a large peak on the excited potential and also the growth of the target population in the ground state (in the lower figure). In conclusion, we have applied an optimal control theory to a model system of the photoisomerization of retinal in order to determine the effects of potential coupling on control mechanisms. The present system has considerably different potential curves from those used previously in studies on the optimal control pathways for curve crossings of small molecules. Optimal pulses that enhance trans–cis isomerization were numerically calculated in two cases of coupling strengths. In the adiabatic case, in addition to a pair of conventional pump and dump subpulses, the optimal pulse includes a shaping subpulse that adjusts the shape of a wave packet through population exchange processes between electronic states. In the medium-coupling case, in which approximately half of the Franck–Condon packet remains in the original potential after passing the curve-crossing region, the calculated pulse directly increases the diabatic coupling using a half-cycle subpulse and achieves similar control to that in the adiabatic case. On the other hand, when such lowfrequency components were prohibited, the opti-
mal pulse designed under this restriction led to a new type of pump–dump control scheme. The restricted optimal pulse creates a shaped packet in the ground electronic state with energy close to the top of the potential barrier that separates the two isomers. This wave packet moves along the adiabatic potential because of its low velocity at the crossing region. At the final step, this packet is transferred to the target state by rapid pump– dump processes, and we have 90% of the population in the target state.
Acknowledgements We are grateful to Professor W. Domcke for his valuable discussions, and the author (Y.O.) also acknowledges stimulating discussions with Professor H. Rabitz. This work was partly supported by a Grant-in-Aid for Scientific Research on Priority Areas, ÔControl of Molecules in Intense Laser FieldsÕ from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of the Japanese Government, and also by a Grant-in-Aid for Scientific Research on Priority Areas (417).
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New quantum control pathway for a coupled-potential system Yukiyoshi Ohtsuki *, Kazuki Ohara, Mayumi Abe, Kazuyuki Nakagami 1, Yuichi Fujimura Department of Chemistry, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan Received 16 August 2002; in final form 26 October 2002
Abstract Optimal control theory is used to design laser pulses that enhance the trans–cis photoisomerization of retinal, which is modeled by coupled potential curves to examine coupling effects on control mechanisms. Based on numerical results, a new type of pump–dump control pathway, in which an optimal pulse achieves high target probability without directly modifying a potential coupling, is proposed. The key feature of this control is that a shaped wave packet passes a curvecrossing point adiabatically in the ground electronic state. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction Non-adiabatic coupling between potential energy surfaces plays an essential role in various photochemical processes, including dissociation, isomerization, electron transfer, and other photochemical reactions [1–3]. It is thus expected that the control of non-adiabatic transitions by means of shaped laser excitation would enable selective induction of subsequent photochemical events with high efficiency. According to this idea, the possibility of controlling branching ratios has been *
Corresponding author. E-mail address: [email protected] (Y. Ohtsuki). 1 Present address: Nippon Timeshare Co. LTD., 3-11-24 Minato-ku, Tokyo 108-8561, Japan.
discussed on the basis of optimal control theory using curve-crossing systems that typically model diatomic molecules [4–10]. Their control pathways were shown to undergo considerable changes depending on the potential shape as well as the control objective. In this Letter, we focus on the trans–cis isomerization of the retinal chromophore in bacteriorhodopsin, which has different potential curves from those studied so far [11]. We show that this isomerization can be controlled by a new type of pump–dump mechanism that utilizes the adiabatic motion of a wave packet in the ground electronic state. It is thought that the photoisomerization of retinal can be described by coupled low-dimensional potentials [11–15], although a thorough understanding of the reaction mechanism has not been attained. This deduction is justified by partial
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(02)02030-4
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Y. Ohtsuki et al. / Chemical Physics Letters 369 (2003) 525–533
dynamic guidance provided by surrounding media, which achieves efficient isomerization. In fact, the potential surfaces relevant to photoisomerization are often modeled by two- or even one-dimensional potentials [11–15], the parameters of which are determined to reproduce experimentally observed spectra. For the sake of comparison, we briefly summarize herein the results of previous theoretical studies in which optimal control pathways for potential crossings of small molecules were examined. Gross et al. [4,5] studied a system with three electronic states, two of which were dissociative electronic excited states coupled with each other. Optimal pulses were designed in order for an initially prepared wave packet to remain or change the original potential after passing a crossing region. They showed that in a diabatic-transition case, an optimal pulse utilizes the optical transitions between the ground and excited states to increase the average energy of a wave packet (the Landau–Zener mechanism). To induce adiabatic transitions, on the other hand, an optimal pulse transfers a wave packet into a third state, and the packet is pumped back to one of the excited states after passing the crossing region. On the other hand, we studied predissociation of NaI using a two-orientation model with the aim of accelerating the predissociation [8,9]. We found two optimal control pathways for this target, i.e., a pump– dump mechanism and the Landau–Zener mechanism, in which excited wave packets are controlled to move along the potential curve adiabatically and diabatically, respectively. Recently, Vivie-RiedelÕs group [10] dealt with the optimal control of a system with a conical intersection and designed a pulse that creates a localized wave packet around the conical intersection. This Letter is organized as follows. In Section 2, we introduce our model potential and coupled pulse design equations. The numerical results and a summary are given in Section 3.
2. Theory We adopt a one-dimensional model for the excited state trans–cis isomerization of a retinal
chromophore in bacteriorhodopsin. This simplified model was proposed by Pollard et al. [11] in order to qualitatively reproduce transient absorption spectra. Here we slightly modify their potential [11] to discuss the effects of non-adiabatic transitions on the laser control of the isomerization. As shown in Fig. 1, we assume two electronic states whose diabatic potential curves, V11 ðhÞ and V22 ðhÞ, are modeled by 50 cm1 harmonic oscillators with minima at dihedral angles of h ¼ 0° and h ¼ 180° corresponding to trans- and cis-isomers, respectively. These two diabatic states interact with each other around the potential crossing whose interaction potential, V12 ðhÞ ½V21 ðhÞ, is approximated by a Gaussian function centered at a crossing point and having a width of Dh ¼ 10°. This molecular system interacts with a timedependent electric field, EðtÞ, through the semiclassical dipole–field interaction. Since it is known that the coherent motion of the excited wave packet is observed during the isomerization process [16], the time evolution of the system is assumed to be described by the Schr€ odinger equation. Within our qualitative treatment of isomerization, this assumption may hold as long as we are concerned with ultrafast (subpicosecond) dynamics. Then, our system obeys the equation of motion:
Fig. 1. Model potential energy curves in an adiabatic representation. A new pump- and dump-control pathway is schematically illustrated by the wave packets and arrows. The solid and dotted arrows indicate optical transitions and wave packet propagation on the ground-state potential, respectively.
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" # o w1 ðh; tÞ ih ot w2 ðh; tÞ " # V12 ðhÞ l12 ðhÞEðtÞ T þ V11 ðhÞ ¼ V21 ðhÞ l21 ðhÞEðtÞ T þ V22 ðhÞ " # w1 ðh; tÞ ; ð1Þ w2 ðh; tÞ where wk ðh; tÞ is the wavefunction of the kth diabatic state (k ¼ 1; 2), and l21 ðhÞ [l12 ðhÞ] is the matrix element of the dipole moment operator, lðhÞ, which is associated with the w1 ! w2 [w2 ! w1 ] transition. They are expressed as l21 ðhÞ ¼ l12 ðhÞ ¼ l0 ðhÞ cos h for convenience. Here we assume that the electric field is linearly polarized. An optimal control pulse is designed according to a standard procedure [17]. A target operator, W, is introduced to specify a physical objective. An optimal pulse is designed so that it maximizes an expectation value of a target operator at a final time, tf , subject to minimum pulse fluence. Then, the optimal pulse is defined by a pulse that gives a maximal value to the objective functional: Z tf 1 2 J ¼ hwðtf ÞjW jwðtf Þi dt½EðtÞ hA 0
Z tf o i t ; 2Re dt nðtÞ þ H wðtÞ ot h 0 ð2Þ wherejwðtÞi and H t denote the total wavefunction and the total Hamiltonian, respectively, the positive constant A, which is called the weight parameter, weighs the significance of the penalty, and jnðtÞi is the total Lagrange multiplier constraining the system to obey the equation of motion, Eq. (1). Applying calculus of variations to Eq. (2), we obtain the following coupled pulse design equations. The optimal pulse is expressed as EðtÞ ¼ A ImfhnðtÞjlðhÞjwðtÞig:
ð3Þ
The time evolution of the total Lagrange multiplier is determined by integrating the equation of motion o i h jnðtÞi ¼ H t jnðtÞi; ð4Þ ot
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with the final condition of jnðtf Þi ¼ W jwðtf Þi:
ð5Þ
Here, the Hamiltonian in Eq. (4), H t , is the total Hamiltonian whose expression is given in Eq. (1).
3. Results and discussion In order to examine the effects of non-adiabatic interaction on an optimal pathway for photoisomerization, we consider two cases, i.e., strong- and medium-coupling cases. We first illustrate these coupling strengths by calculating the time-dependent behavior of a Franck–Condon packet in the absence of electric fields. The Franck–Condon packet is a packet that has the same form as that of the initial state but is on an excited surface, V22 ðhÞ. Throughout this Letter, we assume that the molecular system is initially in the lowest vibrational state of the all-trans isomer. In the strong-coupling (adiabatic) case, 96.5% of the initial population of the Franck–Condon packet adiabatically switches to another diabatic potential after initially passing the crossing point. In the medium-coupling case, on the other hand, 44.5% of the initial population remains in the original diabatic potential, V22 ðhÞ, due to the non-adiabatic interaction. Our purpose here is to design an optimal pulse that transfers as much of the initial state (the lowest vibrational state of the all-trans isomer) as possible to the cis-isomer. For this purpose, we adopt a target operator that projects the subspace spanned by the lowest six vibrational states associated with a cis-isomer, {jcis vi}, W ¼
5 X
jcis vihcis vj;
ð6Þ
v¼0
where the energy eigenvalues of these states are far below the potential barrier that separates the two isomers. Optimal pulses are obtained by solving the pulse design equations together with this target operator and the amplitude parameter, A ¼ 4:0 106 . In simulations shown below, the final time is set to tf ¼ 500 fs, which is shorter than the period of the wave packet oscillation in the excited adiabatic state. This time interval is divided into
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Nt ¼ 100,000 time steps to numerically integrate the equations of motion, Eqs. (1) and (4). We prevent a wave packet from rotating beyond 2p by removing a periodic boundary condition from the potential curves [11]. As a tradeoff, we assume an artificial boundary condition in which a fictitious angle range 3p is used to avoid wave packet reflection at the edge of the grid. Then, adopting 1024 uniformly spaced grid basis, the time evolution of the wave packet is calculated by the secondorder, split-operator method with the fast Fourier transform (FFT) technique [18]. All of the potential couplings are calculated using the Pauli matrix [4]. In the first two numerical examples (Figs. 2 and 3), we assume a constant value of l0 ðhÞ ¼ l0 . For an iteration algorithm, we combine the method proposed by Zhu and Rabitz [19] and that proposed by Tannor et al. [20]. Since the algorithm proposed by Zhu and Rabitz [19] has high nu-
Fig. 3. Results in the medium-coupling case. (a) The optimal control pulse as a function of time and (b) the time evolution of the target probability (solid line) and that of the populations on the ground (dotted line) and excited (dot-dashed line) adiabatic states. The dashed line represents the time evolution of the target probability calculated using the filtered pulse.
Fig. 2. Results in the strong-coupling (adiabatic) case. (a) The optimal control pulse as a function of time and (b) the time evolution of the target probability (solid line) and that of the populations on the ground (dotted line) and excited (dotdashed line) adiabatic states.
merical reliability but sometimes suffers from slow convergence, we use this algorithm for the first 1000 iteration steps, by which the shape of an initial input pulse is adjusted so that it is close to an optimal solution. We then use the Krotov method [20] to save CPU time, since its rough search improves the speed of convergence near the optimal solution. In our calculations, a pulse is regarded as an optimal solution when the difference between the values of J in the final two steps is DJ =J 6 3:9 106 . Fig. 2 shows (a) the optimal pulse and (b) the temporal behavior of the target probability and that of the population on each adiabatic potential in the strong-coupling (adiabatic) case. This optimal pulse transfers 97.2% of the population to the target state [Eq. (6)]. The pulse consists of pump, shaping and dump subpulses, although we cannot
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clearly distinguish them due to the rather temporally broad and overlapped structure of the control pulse. The modulated subpulse in the time interval t 2 ½0 fs; 170 fs transfers the initial population into the electronic excited state, simultaneously adjusting the shape of the excited wave packet. We see from Fig. 2b that the excited packet then moves along the adiabatic potential during t 2 ½170 fs; 290 fs. Note that the total excited(ground-) state population does not change during this time interval, although we see intense pulse sequences (t 230 fs) in Fig. 2a. This means that population exchange processes between the two electronic states must occur. Through these processes, the shape of the wave packet is gradually adjusted, which is appropriate for the following dumping process. Such a wave packet shaping mechanism through population exchange processes was numerically observed in a different system with a different physical objective in our previous study [21]. We thus conclude that wave packet shaping through population exchange processes is essential for achieving a high control yield in a multi-electronic system. Finally, a subpulse during t 2 ½290 fs; 370 fs transfers the wave packet into the target state. The large temporal width of this dump pulse indicates that the wave packet has a low velocity and a spatially broadened structure around the Franck–Condon region accessible from the target state. Next, we consider the medium-coupling case. The calculated pulse and the time evolution of the target probability and that of the population on each adiabatic state are shown in Figs. 3a,b, respectively. This optimal pulse is composed of three well-separated subpulses and transfers 95.6% of the population to the target state. The first subpulse not only transfers the initial population to the electronic excited state but also adjusts the shape of the wave packet. Immediately after the excitation, the excited packet has nearly zero velocity and requires some time to leave the Franck– Condon region, thus providing sufficient time for the pulse to shape the packet. The combination of pump and shaping pulses results in a modulation structure in the first subpulse. The second subpulse may be called a half-cycle pulse. The electric field with a positive amplitude strengthens the diabatic
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potential coupling around the potential crossing (h ¼ 110°) and prevents the excited packet from moving along the diabatic potential V22 ðhÞ. By these first two subpulses, a localized wave packet is created on the diabatic potential V11 ðhÞ around the Franck–Condon region associated with the target state. The third subpulse efficiently transfers this localized packet into the target state specified by the operator in Eq. (6). Since the packet still has finite velocity at this time, the subpulse has to complete the dumping process very quickly, which is the reason why the dump pulse has a sharp temporal width. The feature of the optimal pulse in Fig. 3a is that it includes a low-frequency subpulse (a halfcycle pulse). To determine the role of this subpulse, we calculate the time evolution of the wave packet using a modified pulse that is obtained by eliminating low-frequency components from the original pulse by Fourier filtering. The time-dependent behavior of the expectation value of the target operator is shown by a dashed line in Fig. 3b, in which the target population at the final time is reduced to 52.0%. This means that it is essential to utilize a low-frequency pulse in order to obtain high target probability, as long as we adopt a conventional pump–dump control scheme. However, such a half-cycle pulse with high intensity and definite phase is not experimentally feasible at present, although half-cycle pulses with much lower intensities than that in Fig. 3a can be generated by the several pulse compression techniques relevant to their temporal widths [22,23]. The question is whether there is an alternative control pathway that achieves high target probability without using low-frequency pulses. To answer this question, we design an optimal pulse that does not include low-frequency components. To filter out the low-frequency components from the optimal pulse, Gross et al., for example, proposed a gradient filtering procedure. Here we will adopt an alternative convenient procedure [9] that is valid as long as the so-called frozen wave packet assumption [24,25] is fulfilled during the optical interaction. To outline our method, consider the case where a molecular system is excited by a pulse EðtÞ that has a small temporal width, Dt, and is centered at t ¼ 0. Note that under the frozen wave
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packet assumption [24,25], the kinetic energy operator is ignored in the propagator associated with the molecular Hamiltonian, resulting in the following approximate expression: expðiH Dt= hÞ ’ expðiV Dt= hÞ;
ð7Þ
where Dt is the period of the optical interaction and V is the total molecular potential. Utilizing Eq. (7) and the rotating wave approximation (RWA), the formal solution to Eq. (1), jwI ðtÞi, is expressed in the interaction picture as jwI ðtÞi jw0 i Z 1 h ’ 2Dt dx l21 ðhÞE~ðxÞ expfi½U21 ðhÞ xDtg 1 i þ h:c: jwI ðDtÞi; ð8Þ where jw0 i is the initial state, hU21 ðhÞ ¼ V22 ðhÞ V11 ðhÞ, and E~ðxÞ is a Fourier component of the pulse. When the pulse does not induce rapid Rabi oscillations, the wavefunction in Eq. (8) is a slowly varying function of time (the interaction picture). Thus, only the frequency components that satisfy the relation jV22 ðhÞ V11 ðhÞ hxj 6 h=Dt
ð9Þ
can induce optical transitions; that is, we can restrict frequency components in the optimal pulse by imposing restrictions on the optical transition region through the transition moment function, l0 ðhÞ [9]. According to this filtering procedure, we tentatively replace l0 ðhÞ ¼ l0 with 1 l0 ðhÞ ¼ l0 1 þ exp½aðh h1 Þ
1 þ ; ð10Þ 1 þ exp½aðh h2 Þ where a ¼ 0:5, h1 ¼ hcross 30:0, and h2 ¼ hcross þ 30:0 with the crossing point hcross ¼ 110:0°. Using Eq. (10), we solve the pulse design equations. Note that because l0 ðhÞ ¼ 0 around the potential crossing, this optimal pulse cannot directly modify the potential coupling. Fig. 4 shows (a) the calculated pulse, and (b) the time evolution of the target state and that of the population on each adiabatic potential. In these calculations, we found
Fig. 4. Results in the case where a pulse is designed with restricted frequency components. (a) The optimal control pulse as a function of time and (b) the time evolution of the target probability (solid line) and that of the populations on the ground (dotted line) and excited (dot-dashed line) adiabatic states.
that the pulse shown in Fig. 4a could realize virtually the same control scheme without assuming the artificial transition moment function given by Eq. (10). This was actually confirmed by numerical calculations with use of the original constant transition moment, l0 ðhÞ ¼ l0 . This means that our filtering procedure based on the frozen wave packet assumption offers a convenient alternative to eliminate low-frequency components from the optimal pulse. From Fig. 4a, we can see that the optimal pulse achieves a high isomerization probability of 90.0% without directly controlling the potential coupling but by adopting a considerably different control pathway from that in Fig. 3. The difference in control pathways can be seen by comparing the time-dependent behavior of the populations in Fig. 3b with that in Fig. 4b. In Fig. 3b, the wave
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packet moves on the excited adiabatic potential during the time interval, t 2 ½110 fs; 280 fs, whereas, in Fig. 4b, the packet moves on the ground potential. Based on these features, along with time- and frequency-resolved spectra (not shown here), we deduced the following new control mechanism. Before t 70 fs, the pulse creates a shaped wave packet on the excited adiabatic potential through population exchange processes. This excited packet is then transferred to high vibrational states in the ground state at t 100 fs. The average energy of this packet is close to the
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top of the potential barrier that separates the two isomers. Therefore, when passing the crossing region, the packet tends to move along the adiabatic potential because of its low velocity. After passing the crossing point, the packet continues to move toward the potential well associated with the cisisomer. Immediately before the packet reaches the target region, the pulse around t 300 fs pumps the wave packet up to the excited electronic state again. This packet is immediately transferred to the target state in the ground electronic state around t 350 fs.
Fig. 5. Time evolution of the wave packets in the excited (upper figure) and ground (lower figure) adiabatic states (a) in the case of Fig. 3 and (b) in the case of Fig. 4.
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To more clearly show the new control pathway, the time evolution of the wave packet on each adiabatic potential is presented in Fig. 5. For the sake of comparison, we show time evolutions of wave packets (a) in the case of Fig. 3 and (b) in the case of Fig. 4. In Fig. 5a, we see a typical wave packet motion induced by a pair of conventional pump and dump control pulses, although a modulated structure can also be seen around t 2 ½120 fs; 150 fs corresponding to packet-shaping processes. On the other hand, in the lower figure of Fig. 5b, the shaped wave packet appears around h 50° at t 70 fs in the ground (adiabatic) electronic state. The velocity of this packet gradually decreases as the packet climbs up the potential hill, and after passing the curve-crossing point, the velocity increases again. In the upper figure, we see that a very small portion of the wave packet is transferred to the excited state due to non-adiabatic coupling, which appears and remains around h 110° in the excited state. At the final stage around t ½250 fs; 350 fs, we observe a large peak on the excited potential and also the growth of the target population in the ground state (in the lower figure). In conclusion, we have applied an optimal control theory to a model system of the photoisomerization of retinal in order to determine the effects of potential coupling on control mechanisms. The present system has considerably different potential curves from those used previously in studies on the optimal control pathways for curve crossings of small molecules. Optimal pulses that enhance trans–cis isomerization were numerically calculated in two cases of coupling strengths. In the adiabatic case, in addition to a pair of conventional pump and dump subpulses, the optimal pulse includes a shaping subpulse that adjusts the shape of a wave packet through population exchange processes between electronic states. In the medium-coupling case, in which approximately half of the Franck–Condon packet remains in the original potential after passing the curve-crossing region, the calculated pulse directly increases the diabatic coupling using a half-cycle subpulse and achieves similar control to that in the adiabatic case. On the other hand, when such lowfrequency components were prohibited, the opti-
mal pulse designed under this restriction led to a new type of pump–dump control scheme. The restricted optimal pulse creates a shaped packet in the ground electronic state with energy close to the top of the potential barrier that separates the two isomers. This wave packet moves along the adiabatic potential because of its low velocity at the crossing region. At the final step, this packet is transferred to the target state by rapid pump– dump processes, and we have 90% of the population in the target state.
Acknowledgements We are grateful to Professor W. Domcke for his valuable discussions, and the author (Y.O.) also acknowledges stimulating discussions with Professor H. Rabitz. This work was partly supported by a Grant-in-Aid for Scientific Research on Priority Areas, ÔControl of Molecules in Intense Laser FieldsÕ from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of the Japanese Government, and also by a Grant-in-Aid for Scientific Research on Priority Areas (417).
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