Optimal Control Simulation of Field-Free Molecular Orientation

Article pubs.acs.org/JPCA

Optimal Control Simulation of Field-Free Molecular Orientation: Alignment-Enhanced Molecular Orientation Katsuhiro Nakajima, Hiroya Abe, and Yukiyoshi Ohtsuki* Department of Chemistry, Graduate School of Science, Tohoku University, Sendai 980-8578, Japan ABSTRACT: Nonresonant optimal control simulation is applied to a CO molecule to design two-color phase-locked laser pulses (800 nm + 400 nm) with the aim of orienting the molecule under the field-free condition. The optimal pulse consists of two subpulses: the first subpulse aligns the molecule and the second one orients it. The molecular alignment induced by the first subpulse considerably enhances the degree of orientation, the value of which is close to an ideal value at temperature T = 0 K. To confirm the effectiveness of this alignment-enhanced orientation mechanism, we adopt a set of model Gaussian pulses and calculate the maximum degrees of orientation as a function of the delay time and the intensity. In finite-temperature (T = 3.0 K and T = 5.0 K) cases, although the alignment subpulse can improve the degree of orientation, the control achievement decreases with temperature rapidly; this decrease can be attributed to the initial-state-dependent (phase-shifted) rotational wave packet motion. violating interaction is utilized.29−36 Examples include theoretical proposals that use (half-cycle) terahertz pulses, some of which combine terahertz excitation with nonresonant laser pulses.29−32 Although the use of the terahertz pulses shows promise, there still remain technical problems in the generation of intense terahertz pulses. Since the early study by Friedrich and Herschbach,33 the combination of static fields with the excitation by nonresonant laser pulses is known to achieve reasonably high degrees of orientation.14,34−36 The control in the presence of a static field, however, does not result in completely field-free molecular orientation. In the second category, an intense two-color phase-locked laser pulse, which typically consists of 800 and 400 nm components, is used.37−41 The 400 nm component is usually generated by the sum frequency generation of the 800 nm component. The parity-violating interaction is introduced by the hyperpolarizability. Some experimental studies demonstrated the effectiveness of this scheme by employing simple molecules, such as CO40 and OCS.41 However, the observed degrees of orientation are not so high. This limited success is mainly attributed to weak hyperpolarizability interactions, which also explains the requisite intense laser pulses used in the experiments. In light of advanced pulse shaping techniques in the 800 nm region, we may expect to achieve a reasonably high degree of orientation with a shaped laser pulse with mildly high intensity. From this viewpoint, Lapert et al.42 numerically designed two-color laser pulses on the basis of their optimal

I. INTRODUCTION The control of molecular alignment/orientation has attracted considerable interest for the past few decades as it has enabled molecular processes to be induced in a molecular-fixed frame. When a molecule is in its ground electronic state, a torque required for its alignment/orientation control is generated through dipole and/or induced-dipole interactions.1,2 To avoid possible side effects due to the presence of an electric field, molecular alignment/orientation under the field-free condition is favorable. As regards alignment control, impulsive Raman excitation of rotational states with a nonresonant laser pulse is often used for this purpose,3−16 in which an aligned state appears regularly (rotational revival) after the pulse.17 Because a single pulse excitation has some inherent limitations that hinders the improvement of the degree of alignment,7,14 various effective multipulse schemes have been reported. In addition to aligning a molecule in a specific direction (1D alignment), there have been several studies on 3D alignment by using laser pulses with several polarization conditions and by a set of linearly polarized pulses.18−22 Today, alignment control studies have shifted their focus from the elucidation of basic mechanisms to optimization as well as the wide range of applications.23−26 Examples of the latter include molecular orbital tomography24,25 and time-resolved imaging of a chemical reaction.26 For asymmetric molecules, molecular alignment control would be insufficient to realize molecular-fixed-frame experiments because the aligned molecules are regarded as a mixture of molecules with opposite head-versus-tail orders.27−41 The primary difference between alignment control and orientation control is that the latter requires parity-violating interactions. From the viewpoint of parity-violating interactions, orientation control can be classified into two categories. In the first category, a dipole moment that leads to the lowest-order parity© 2012 American Chemical Society

Special Issue: Jörn Manz Festschrift Received: May 29, 2012 Revised: July 26, 2012 Published: July 27, 2012 11219

dx.doi.org/10.1021/jp3052054 | J. Phys. Chem. A 2012, 116, 11219−11227

The Journal of Physical Chemistry A

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The phase, η, determines the degree of asymmetry of the optical interactions. For example, η = π/2 provides no asymmetry, which results in no control of orientation. Throughout the present study, we use a fixed value of η = 0. The dynamical behavior of the molecular system is described by the time-evolution operator, U(t,0), that obeys the equation of motion:

control method. In one of their simulations, they found a nonintuitive solution, i.e., the parity-violating (asymmetric) interaction followed by a parity-conserving (symmetric) interaction. The present study focuses on the second category, i.e., control with two-color phase-locked laser pulses. Through a case study of a CO molecule, we explore an effective way to realize a high degree of orientation. To this end, we apply our own optimal control algorithm to the design of an orientation pulse.43−45 As it is known that there exist multiple solutions in optimal pulse design,46,47 our algorithm can find an alternative solution with effectiveness similar to that reported by Lapert et al.42 In fact, we will propose an alignment-enhanced molecular orientation mechanism that will be derived from our optimal control simulations. In section II, we briefly summarize the procedure of our optimal control simulation. The algorithmic details are given in the Appendix. In section III, numerical results are presented.

iℏ

2

|ψ (t )⟩ = U (t ,0)|ψ0⟩

F = ⟨ψ (tf )|W |ψ (tf )⟩

(8)

For the sake of molecular orientation, the natural choice would be Jmax

W = P cos θP

with

P=

J

∑ ∑

|JM ⟩⟨JM |

J = 0 M =−J

(9)

where the projector, P, is introduced to restrict the number of rotational states to be optically controlled. This is because controlling the highly excited rotational states would not be experimentally realistic or feasible. By applying calculus of variations under the constraint of eq 7, δF δF = =0 δεω(t ) δε2ω(t )

(1)

(9)

we obtain the optimality conditions:

where α(θ ) = (α − α⊥) cos2 θ + α⊥

(7)

In the optimal control simulation, we first choose a target operator, W, to specify a physical objective. The optimal pulse is designed so that it gives the maximum expectation value at a specified time, tf:

1 α(θ )[E(t )]2 2

1 − β(θ )[E(t )]3 6

(6)

When the rotor is initially in the lowest state, |ψ0⟩ = |J = 0 M = 0⟩, the wave function at time, t, is given by

II. OPTIMAL CONTROL SIMULATION We consider a CO molecule modeled by a rigid rotor, which is specified by a rotational constant, B, and an angular momentum operator, J.̂ The molecule is assumed to interact with a linearly polarized laser pulse, E(t), through a permanent dipole moment, μ, polarizability components, α∥ and α⊥, and hyperpolarizability components, β∥ and β⊥, where indices ∥ and ⊥ mean the components parallel and perpendicular to the molecular axis, respectively. Let θ be the angle between the polarization vector of the laser pulse and the molecular axis. Then, the total Hamiltonian is expressed as H t = BJ ̂ − μE(t ) cos θ −

∂ U (t ,0) = H̅ tU (t ,0) ∂t

⎧ ⎫ ⎡1 ⎤ 1 Im⎨⟨ξ(t )|⎢ α(θ) εω(t ) + β(θ) εω(t ) ε2ω(t )⎥|ψ (t )⟩⎬ = 0 ⎣2 ⎦ ⎩ ⎭ 4

(2)

(10)

and β(θ ) = (β − 3β⊥) cos3 θ + 3β⊥ cos θ

and ⎧ ⎫ ⎡1 ⎤ 1 Im⎨⟨ξ(t )|⎢ α(θ ) ε2ω(t ) + β(θ ) εω2(t )⎥|ψ (t )⟩⎬ = 0 ⎣ ⎦ ⎩ ⎭ 2 8

(3)

The first term of the right-hand-side of eq 1 is the rigid-rotor Hamiltonian, the eigenstates of which are the spherical harmonic wave functions denoted by {|J M⟩}. The magnetic quantum number, M, is conserved during the optical transitions because the laser field is linearly polarized. In the present study, we consider a two-color laser field E(t ) = εω(t ) cos(ωt ) + ε2ω(t ) cos(2ωt + η)

(11)

where the Lagrange multiplier that represents the constraint is expressed as |ξ(t )⟩ = U (t ,tf )|ξ(tf )⟩ = U (t ,tf )W |ψ (tf )⟩

Equations 7 and 10−12 are composed of the pulse design equations. We solve the nonlinear equations by adopting the iteration algorithms that guarantee monotonic convergence (Appendix). In the numerical calculations, we adopt the rotational constant, B = 1.92 cm−1,48 corresponding to the rotational period, Trot = 1/(2cB) = 8.68 ps, with c being the velocity of light. The values of the polarizability components are taken from ref 49 so that α∥ = 15.63 au, α⊥ = 11.97 au, β∥ = 30.0 au, and β⊥ = 8.4 au. All the wave functions and their associated Lagrange multipliers are expanded in terms of the rotational states {|J M⟩} up to J = 17. The time evolution of the expansion coefficients is calculated by the fifth-order Runge−Kutta method, in which the temporal grid is set to 2Trot × 10−6.

(4)

that contains the two frequency components specified by ω and 2ω with the relative phase, η. In eq 4, εω(t) and ε2ω(t) are the envelope functions. If we further assume that ω and 2ω are much higher than those associated with the rotational transitions, we can take the cycle average over the optical frequencies of the laser field. The cycle-averaged Hamiltonian is expressed as 2

H̅ t = BJ ̂ − −

1 α(θ )[εω 2(t ) + ε2ω 2(t )] 4

1 β(θ )εω 2(t ) ε2ω(t ) cos η 8

(12)

(5) 11220



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Throughout the present study, the degree of orientation and that of alignment are evaluated by the expectation values of cos θ and cos2 θ, respectively.

III. RESULTS AND DISCUSSION A. Optimal Control Simulation in the Zero-Temperature Case. In the zero-temperature case, the initial state is given by |ψ0⟩ = |J = 0 M = 0⟩. As the magnetic quantum number is conserved, the wave function is expressed in terms of the set of eigenstates, {|J M = 0⟩}. We choose Jmax = 6 to specify the control target in eq 9. We then diagonalize the operator, cos θ, by using the set of rotational states, {|J M = 0⟩; J = 0, ..., Jmax = 6}, to obtain the eigenstate, |χ⟩, with the maximum eigenvalue, 0.949. The optimal pulse is designed to maximize the expectation value, F = ⟨ψ(tf)|W|ψ(tf)⟩, with W = |χ⟩⟨χ| at a specified final time tf = Trot. Figure 1 shows the convergence Figure 2. (a) Envelope functions, εω(t) (red) and ε2ω(t) (blue), of the optimal orientation pulse (see eq 4) as a function of time when the relative phase η = 0. Time is measured in units of the rotational period, Trot. (b) Degree of orientation, ⟨cos θ⟩(t) = ⟨ψ(t)|cos θ|ψ(t)⟩, and that of orientation, ⟨cos2 θ⟩(t) = ⟨ψ(t)|cos2 θ|ψ(t)⟩, are presented by the red-solid and blue-dashed lines, respectively. (c) The population in the even-J states and that in the odd-J states are shown by black dashed and solid lines, respectively. The purple line shows the time-dependent average quantum number, the values of which are provided on the right ordinate. The two vertical dotted lines serve as a guide for the eyes.

the orientation, we consider only the two major subpulses to examine the control mechanisms below. The role of the first subpulse is to create an aligned wave packet. As shown in Figure 2c, the average value of the rotational quantum number rapidly increases although no excitation of the rotational states with odd quantum numbers (i.e., odd-J states) can be seen. The second subpulse excites the odd-J states that are essential to realize the orientation. This result suggests a “two-step” control mechanism in which the control is achieved by a combination of an “alignment” pulse and an “orientation” pulse. According to this simplified picture, first, an alignment pulse creates an aligned wave packet that equally contains the components oriented along one direction and those along the opposite direction. When an orientation pulse is applied to this wave packet, one of the components would be stabilized whereas the other components would become energetically destabilized and start moving toward the opposite direction. These two components would meet and cause constructive interferences to realize a high degree of orientation. In this sense, the twostep mechanism could be referred to as an alignment-enhanced orientation and, thus, we will use this phrase in the following. If we take a closer look at the structure of the second subpulse, we find that it is characterized by a dip, the timing of which corresponds to the maximum expectation value of ⟨J⟩(t) (Figure 2c); that is, the first half of the second subpulse (i.e., before the dip) excites the rotational states and the second half (i.e., after the dip) induces stimulated emission to excite the odd-J states. Through the excitation and/or redistribution of the population, the population in the even-J states and that in the odd-J states are adjusted to have almost the same value (Figure 2c). If the above-mentioned two-step mechanism could achieve a high degree of orientation, the set of optimal subpulses would be mimicked by a set of Gaussian pulses, in which the

Figure 1. Convergence behavior as a function of the number of iteration steps is shown by (a) the values of the kth objective functional, F(k) = | ⟨χ|ψ(k)(tf)⟩|2 (see text) and (b) the values of the difference in objective functional between adjacent iteration steps, δF(k) = F(k) − F(k−1).

behavior as a function of the number of iteration steps when the search parameters (Appendix) are constant, λ1(t) = λ2(t) = 2.5 × 1038. The initial trial field is chosen as ⎡ (t − t /2)2 ⎤ f ⎥ εω(0)(t ) = ε2(0) ω (t ) = ε0 exp⎢ − 2σ 2 ⎦ ⎣ ⎧ sin(πt /2ζ ) (0 ≤ t ≤ ζ ) ⎪ ⎪ (ζ ≤ t ≤ t f − ζ ) × ⎨1 ⎪ ⎪ sin[π (t − t )/2ζ ] (t − ζ ≤ t ≤ t ) ⎩ f f f

(13)

with ε0 = 4.0 GV/m, σ = 1.5 ps, and ζ = 0.15 Trot. It clearly shows the monotonic convergence of the present algorithm as well as its numerical accuracy. In fact, the converged value of F = 0.978 is quite high (the ideal value Fideal = 1.00), which leads to the degree of orientation, ⟨cos θ⟩(tf) = ⟨ψ(tf)|cos θ|ψ(tf)⟩ = 0.930 (the ideal value is 0.949). Figure 2 summarizes the results of the optimal control simulation. The optimal pulse in Figure 2a is composed of two intense subpulses together with several weak subpulses. We see from Figure 2b that the optimal pulse creates a highly oriented wave packet at the specified final time. If we remove the weak subpulses, e.g., after t ≃ 0.8 Trot in Figure 2a, the degree of orientation slightly decreases in value from 0.930 to 0.922. As we have confirmed that the weak subpulses contribute little to 11221



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confirms the effectiveness of the alignment-enhanced orientation mechanism. For reference, in the absence of the alignment pulse, the orientation pulse leads to the maximum degree of orientation, 0.417. We also note that for negative values of the delay time, τ, i.e., when the orientation pulse appears before the alignment pulse, we could not find the regions with high degrees of orientation (not shown). As this is in contrast with the mechanism proposed by Lapert et al.,42 we can say that we have found a different optimal solution from theirs. To see the details, we show the cuts along the dotted lines (a) and (b) in Figures 4 and 5, respectively. Figure 4 shows the

alignment (first) and orientation (second) subpulses are approximated by ⎡ t2 ⎤ Ealignment(t ) = εa exp⎢ − 2 ⎥ cos(ωt ) ⎣ 2σa ⎦

(14)

and ⎡ (t − τ )2 ⎤ ⎥ cos[ω(t − τ )] Eorientation(t ) = εω exp⎢ − 2σo 2 ⎦ ⎣ ⎡ (t − τ )2 ⎤ ⎥ cos[2ω(t − τ )] + ε2ω exp⎢ − 2σo 2 ⎦ ⎣ (15)

respectively. Here, we employ the fixed values of εa = 13.8 GV/ m, ε2ω = (11.3/14.0)εω, σa = 110 fs, and σo = 240 fs, which are estimated from the optimal pulse in Figure 2a. The peak intensity of the alignment pulse (eq 14) is 2.5 × 1013 W/cm2, which is about one-fifth of that used in the orientation experiment.40 The wavelength associated with the optical frequency, ω, is set to 800 nm. Note that the following numerical results do not depend on the choice of the frequencies as long as they are much higher than those of the rotational transitions. By using these Gaussian alignment and orientation pulses, we calculate the maximum degrees of orientation, ⟨cos θ⟩max, as a function of parameters τ and εω. To ensure that we find the maximum value for a given set of Gaussian pulses, we numerically integrate the equation of motion at least for one rotational period after the second pulse because the rotational wave packet of a rotor is known to show the same degree of orientation at every rotational period, Trot (revival). Figure 3 shows the contour plot of the maximum degrees of orientation, ⟨cos θ⟩max, as a function of the delay time, τ, and the amplitude, εω, in which the position corresponding to the optimal solution is marked by a cross. In Figure 3, we see several regions with high values that appear around εω = 14.0 GV/m and periodically along the delay time. The highest value of 0.888 in Figure 3, which is located close to the cross,

Figure 4. Red line: maximum degrees of orientation as a function of the amplitude, εω, with a fixed value of the delay time, τ = 0.49 Trot (4.25 ps), which corresponds to the cut along the dotted line (a) in Figure 3. The population in the even-J states and that in the odd-J states are represented by black dotted and solid lines, respectively.

Figure 5. Red line: maximum degrees of orientation as a function of the delay time (measured in units of Trot) with a fixed value of the amplitude, εω = 14.0 GV/m, which corresponds to the cut along the dotted line (b) in Figure 3. Blue dashed line: degree of alignment at the timing of the peak of each orientation pulse.

maximum degrees of orientation as a function of the amplitude, εω, for a fixed value of τ = 0.49Trot(4.25 ps), in which the population in the even-J states (dotted line) and that in the odd-J states (solid line) are also presented. All the lines in Figure 4 show oscillating behavior as a function of εω. Their oscillation periods become shorter as εω increases because a larger value of εω leads to a larger Rabi frequency. Figure 4 suggests that a larger amplitude of the orientation pulse does not necessarily lead to a higher degree of orientation. The amplitudes associated with the high degrees of orientation are an indication that the even-numbered-state population is equal to the odd-numbered-state population. This could be understood by the fact that the interference between the even-J and odd-J states are essential in the orientation control. For more quantitative discussion, we expand the wave function in eq 7 in

Figure 3. Contour plot of the maximum degrees of orientation, ⟨cos θ⟩max, as a function of the delay time, τ, and the amplitude, εω, of the set of Gaussian pulses in eqs 14 and 15. For each set of Gaussian pulses, we calculate the time evolution at least for one rotational period after the second pulse to ensure that we find ⟨cos θ⟩max (also see text). Time is measured in units of Trot. The two dotted lines orthogonal to each other are introduced for the discussion in Figures 4 and 5. 11222



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(approximately) equally populated; that is, the εω:ε2ω = 1:1 mixture of the frequency components is not an optimal orientation pulse. At the same time, we could say that the degree of orientation is not so sensitive to the ratio compared to the other parameters. such as amplitude (Figure 4) and delay time (Figure 5). In the above discussion, the pulse parameters are chosen so that they are consistent with the optimal pulse in Figure 2a. To further illustrate the effectiveness of the alignment-enhanced orientation, we calculate the degree of orientation with another set of pulse parameters. In this example, the Gaussian pulses are characterized by narrower temporal widths and slightly higher intensities, i.e., the temporal widths and amplitudes are set to σa = σo = 50 fs, εa = 15.0 GV/m, and εω = ε2ω = 20.0 GV/m, respectively (see eqs 14 and 15). Figure 7 shows the maximum

terms of the rotational states {|J M = 0⟩} and calculate the expectation value of cos θ, i.e., the degree of orientation ⟨ψ (t )|cos θ|ψ (t )⟩ =

∑ {C*J+ 1⟨J + 1 M = 0|cos θ|J M = 0⟩CJ + cc} J=0

(16)

where CJ ≡ ⟨J M = 0|ψ(t)⟩. The conditions for {CJ} to give the maximum degree of orientation are derived by differential calculus subject to the normalization condition. Because all the equations involved in this maximization problem are symmetric with respect to the even-J and odd-J states, a high degree of orientation is associated with the equal population of the even-J and odd-J states. It explains the reason why the maximum degree of orientation semiquantitatively coincides with the equal population as shown in Figure 4. Figure 5 shows the maximum degrees of orientation as a function of the delay time, τ, for a fixed value of εω = 14.0 GV/ m. For reference, we also plot the degree of alignment at the timing of the peak of each orientation pulse. In general, the high degrees of orientation well coincide with the high degrees of alignment; that is, when the highly aligned rotational wave packet is excited by the orientation pulse, a high degree of orientation tends to appear. It is worth noting that at τ ≃ 0.07Trot, τ ≃ 0.35Trot, and τ ≃ 0.63Trot, the high degrees of orientation approximately coincide with the timing that corresponds to the decrease of the degrees of alignment with large velocity (in the sense of classical mechanics). This suggests that there may be two kinds of alignment-enhanced orientation mechanisms. So far, we have assumed a fixed ratio of the amplitudes, ε2ω = (11.3/14.0)εω, in the orientation (two-color) pulse. In Figure 6,

Figure 7. Red line: maximum degrees of orientation as a function of the delay time (measured in units of Trot) with fixed values of the amplitudes (see the text). Blue dashed line: degree of alignment at the timing of the peak of each orientation pulse.

degrees of orientation as a function of the delay time and the degrees of alignment at the timing when the orientation pulse is applied. Similarly to the results in Figure 5, we see a good coincidence of the two degrees of control, which can confirm the effectiveness of the combination of the alignment and orientation pulses. In Figure 7, the alignment pulse creates a superposition state that mainly consists of J = 0 and J = 2 states, resulting in a simpler behavior of the degree of alignment than that in Figure 5. Actually, in Figure 5, the aligned wave packet is composed of the rotational states with quantum numbers of up to ca. J = 4. B. Alignment-Enhanced Molecular Orientation in Finite-Temperature Cases. In this subsection, we extend the alignment-enhanced orientation scheme to finite-temperature cases. To evaluate the control performance, we first estimate the “ideal” highest degree of orientation that would be realized by a specified maximum rotational state, Jmax, at a specified temperature. For this purpose, we first solve the eigenvalue problem for each magnetic quantum number, M,

Figure 6. Red solid line: maximum degrees of orientation as a function of the amplitude ratio, εω/ε2ω. The delay time is set to τ = 0.49 Trot (4.25 ps). Black solid and dashed lines: population in the even-J states and that in the odd-J states, respectively. The normalized value of εω2ε2ω is represented by the red dotted line.

we show the maximum degree of orientation as a function of the ratio, εω/ε2ω, in which the total energy of the orientation pulse is set to the same value as that used in Figure 3. As for the delay time, we assume a fixed value of τ = 0.49 Trot. In Figure 6, we also show the population in the even-J states (black dashed line), that in the odd-J states (black solid line), and the product of the amplitudes associated with the hyperpolarizability interaction, εω2ε2ω (thin red dashed line), the maximum value of which is adjusted to 1.0 for illustrative purposes. Within the parameter values considered here, the highest degree of orientation is achieved when the ratio is adjusted for εω2ε2ω to have the maximum value and the even-J and odd-J states are

cos θ|χn(M) ⟩ = Θ(nM)|χn(M) ⟩

with

Jmax

|χn(M) ⟩ =

∑ |J M⟩⟨J M|χn(M) ⟩ J=0

(17)

with the eigenvalues, Θ(M) ≥ Θ(M) ≥ Θ(M) ≥ .... If we take into 0 1 2 account the restriction due to entropy, the ideal value at temperature, T, is calculated by 11223


The Journal of Physical Chemistry A Jmax

cos θ

ideal

=

J

∑ pJ (T ) ∑ J=0

Article

Θ(JM−|)M|

M =−J

(18)

where pJ(T) is the probability that the rotor is thermally excited to the Jth state.50 Figure 8 shows the estimation. We consider

Figure 8. Ideal value of the degree of orientation as a function of temperature for the given maximum rotational state (see eqs 17 and 18).

the temperature at which the quantum numbers of thermally excited states are well below the specified Jmax. In a lowtemperature region, e.g., T ≤ 5 K, the degree of orientation would have a value larger than 0.7 even in the Jmax = 4 case. At temperature T ∼ 30 K, it can have a value of ∼0.7 if we fully control the rotational states with quantum numbers of up to Jmax = 10; that is, Figure 8 may suggest that a reasonably high degree of orientation would be realized even at finite temperature. To see whether this “optimistic” scenario works or not, we numerically solve the quantum Liouville equation with the set of Gaussian pulses given in eqs 14 and 15. At temperature, T, the density operator is expressed as ρ(t ) = U (t ,0)ρ0 U†(t ,0) =

Figure 9. Contour plots of the maximum degrees of orientation, ⟨cos θ⟩max (see the captions of Figure 3), as a function of the delay time, τ, and the amplitude, εω, of the set of Gaussian pulses in eqs 14 and 15 at (a) T = 3 K and (b) T = 5 K. Time is measured in units of Trot. The dotted line in each figure is introduced for the discussion in Figure 10.

∑ |ψJM(t )⟩pJ (T )⟨ψJM(t )| JM

(19)

with |ψJ M(t)⟩ = U(t,0)|J M⟩, where the time evolution operator is defined by eq 6. In Figure 9, the maximum degrees of orientation as a function of the delay time, τ, and the amplitude, εω, are shown as contour plots at (a) 3.0 K and (b) 5.0 K. The definitions of these properties are the same as those in Figure 3. At 3.0 K (5.0 K), the initial population in the lowest state, |J = 0 M = 0⟩, is reduced to 0.669 (0.458) due to the multiplicities associated with the magnetic quantum numbers. Similarly to Figure 3, the regions with high values tend to appear around εω = 14.0 GV/ m and periodically along the delay time. The highest values are 0.626 and 0.516 in the (a) T = 3.0 K and (b) T = 5.0 K cases, respectively. These highest values are located in the regions associated with the optimal conditions (see the cross in Figure 3). Within the temperature ranges considered here, the results in Figure 3 (T = 0 K) and Figure 9 (finite temperature) share several common features. However, we can see several differences as well. One of the discrepancies is the difference in the number of regions with high values. The number of regions as well as their peak values decrease rapidly as temperature increases. To see the decrease in greater detail, we show the cuts along the fixed amplitude, εω = 14.0 GV/m, i.e., the maximum degrees of orientation as a function of τ in Figure 10 (a) T =

Figure 10. (a) Cut along the dotted line in Figure 9a and (b) that in Figure 9b. In each figure, the red line shows the maximum degrees of orientation as a function of the delay time (measured in units of Trot). The blue dashed line shows the degree of alignment at the timing of the peak of each orientation pulse. For reference, the degrees of orientation in the absence of the alignment pulse are indicated by the black thin-solid lines, (a) 0.194 and (b) 0.122.

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3.0 K and (b) T = 5.0 K. The degrees of alignment at the timing of the orientation-pulse peaks are also plotted. For convenience, the degrees of orientation achieved without the alignment pulse are shown by the horizontal thin solid lines, i.e., 0.194 and 0.122 in Figure 10a,b, respectively. We see that the alignment pulse considerably improves the degree of orientation in both cases. Roughly speaking, high values are obtained when the orientation pulse is applied to the highly aligned wave packet; that is, an increase in the degree of alignment leads to an improvement of the degree of orientation, although this is not always the case. The most prominent exceptional cases appear at the delay time, τ ∼ 0.5 Trot, in which the alignment pulse makes virtually no contribution to the improvement of the degree of orientation (Figure 10a) and even makes worse it (Figure 10b). The time evolution of the degree of orientation associated with the latter case (T = 5 K and τ ∼ 0.5 Trot) is shown in Figure 11. The bold solid (dotted) line shows the component,

numerical simulation with a set of Gaussian pulses. Our numerical findings in the T = 0 K case are as follows: (1) High degrees of orientation appear periodically as a function of the delay time, which coincide with the timings of the high degree of alignment. (2) A more intense orientation pulse does not necessarily lead to a higher degree of orientation in the present control scheme. (The optimal intensity results in almost the same population in the even-J rotational states and the odd-J states.) (3) The degree of orientation is not sensitive to the amplitude ratio of the 800 nm component to the 400 nm component in the orientation pulse. In the finite-temperature (T = 3.0 K and T = 5.0 K) cases, although the alignment pulse can improve control achievement, the degree of orientation decreases with temperature rapidly. The primary reason for this decrease is attributed to the cancellation due to opposite orientation directions that depend on the initial states.

■

APPENDIX: SUMMARY OF MONOTONICALLY CONVERGENT ALGORITHM

Coupled pulse-design equations are solved with the iterative procedure developed in our previous study.43,45 The envelope functions of the laser field, εω(t) and ε2ω(t), are artificially divided into two components so that εω (1)(t) and εω (2)(t), and ε2ω (1)(t) ε2ω (2)(t), respectively. Then, all the equations are rewritten as a symmetrical sum of products of these artificially divided components. We start the iteration with the initial trial field given in eq 13. Because of the flexibility of the algorithm, each iteration step can be represented in several forms. Here, we show one of the examples we actually used in the present study. Each iteration step consists of two auxiliary steps, in which the first (second) auxiliary step calculates the electric field associated with εω(t) [ε2ω(t)]. The algorithm at the kth iteration step is summarized as follows. Ausiliary step 1:

Figure 11. Outside the box, the envelope functions of the 800 and 400 nm pulse components are illustrated by the red and blue lines, respectively. The first pulse (second) pulse is the alignment (orientation) pulse. Inside the box, the red line shows the time evolution of the degree of orientation at T = 5.0 K when the amplitude and the delay time are set to εω = 14.0 GV/m and τ = 0.5 Trot, respectively. The contribution, the initial state of which is the J = 0 (J = 1) state, is represented by the black solid (dashed) line. Time is measured in units of Trot.

the initial condition of which is the J = 0 (J = 1) state. If we focus on each component, we find that the alignment pulse improves the degree of orientation. However, the relative phase of the two components is opposite, which results in the cancellation of the degree of orientation between them and leads to a poor degree of orientation as a whole. This “cancellation” behavior is consistent with a recent numerical study51 in which the authors pointed out that the directions of the induced orientation of a molecule with small hyperpolarizability become opposite depending on the odd or even quantum number of the initial rotational state. The initial phase of the wave packet will be explained by the fact that the wave packet keeps the isotropic distribution just after the pulsed excitation because the rotational motion cannot respond quickly.

iℏ

∂ (k) |ξ ̅ (t )⟩ ∂t 1 ) − 1) α(θ)[εω(k(1) (t ) εω(k(2) (t ) + ε2(ωk −(1)1)(t ) ε2(ωk −(2)1)(t )] 4 ε2(ωk −(1)1)(t ) + ε2(ωk −(2)1)(t ) ⎫ ⎪ (k) 1 − 1) ) ⎬|ξ ̅ (t )⟩ − β(θ ) εω(k(1) (t ) εω(k(2) (t ) ⎪ 8 2 ⎭ (A1) =

{BJ ̂

2

−

with the final condition |ξ̅(k)(tf)⟩ = W|ψ(k−1)(tf)⟩, and iℏ

∂ (k) |ψ̅ (t )⟩ ∂t 1 ) ) α(θ)[εω(k(1) (t ) εω(k(2) (t ) + ε2(ωk −(1)1)(t ) ε2(ωk −(2)1)(t )] 4 ε2(ωk −(1)1)(t ) + ε2(ωk −(2)1)(t ) ⎫ ⎪ 1 ) ) ⎬|ψ̅ (k)(t )⟩ − β(θ ) εω(k(1) (t ) εω(k(2) (t ) ⎪ 8 2 ⎭ =

IV. SUMMARY We have designed a two-color optimal pulse to orient a CO molecule at T = 0 K. The optimal pulse consists of two subpulses. The first subpulse aligns the molecule and the second one is applied to the aligned wave packet to control the orientation. This “two-step” control mechanism, i.e., alignmentenhanced molecular orientation, has been confirmed by

{

2

BJ ̂ −

(A2)

with the initial condition |ψ̅ (0)⟩ = |ψ0⟩. In eqs A1 and A2, the symmetrically divided electric fields are expressed as (k)

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To prove the monotonic convergence, we consider the difference in the objective functional between the kth and (k − (k) 1)th iteration steps, F(k,k−1) = δf(k) 1 + δf 2 , where

− 1) ) (t ) − εω(k(1) (t ) εω(k(1)

⎧ ⎪ (k) − 1) (t ) = −λ1(t )⟨ξ ̅ (t )|⎨2α(θ ) εω(k(2) ⎪ ⎩ − 1) (t ) + β(θ ) εω(k(2)

δf1(k) = ψ̅ (k)(tf )|W |ψ̅ (k)(tf ) − ψ (k − 1)(tf )|W |ψ (k − 1)(tf )

ε2(ωk −(1)1)(t ) + ε2(ωk −(2)1)(t ) ⎫ ⎪ ⎬|ψ (k − 1)(t )⟩ ⎪ 2 ⎭

(A9)

and δf 2(k) = ψ (k)(tf )|W |ψ (k)(tf ) − ψ̅ (k)(tf )|W |ψ̅ (k)(tf )

(A3)

and

(A10)

) (t ) εω(k(2)

−

− 1) (t ) εω(k(2)

For example,

⎧ ⎪ (k) ) (t ) = −λ1(t )⟨ξ ̅ (t )|⎨2α(θ ) εω(k(1) ⎪ ⎩ + β (θ )

ε2(ωk −(1)1)(t ) ) (t ) εω(k(1)

(A11)

where |δψ̅ (tf)⟩ = |ψ̅ (tf)⟩ − |ψ (tf)⟩ and P1(t) = ⟨ξ̅k(t)|δψ̅ (k,k−1)(t)⟩. If we differentiate P1(t) with respect to time, we have (k,k−1)

+ ε2(ωk −(2)1)(t ) ⎫ ⎪ ⎬|ψ̅ (k)(t )⟩ ⎪ 2 ⎭

) (t ) + β(θ) εω(k(1)

∂ (k) |ξ (t )⟩ ∂t 2

−

(A5)

−

2Re{P1(tf )} =

(A6)

■

dt

1 − 1) {[ε(k) (t ) − εω(k(1) (t )]2 4ℏλ1(t ) ω(1) (A13)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected].

= −λ 2(t )⟨ξ(k)(t )|{4α(θ ) ε2(ωk −(2)1)(t )

Notes

The authors declare no competing financial interest.

(A7)

■

and

ACKNOWLEDGMENTS We thank Professor Jörn Manz for stimulating discussions in Sendai and for helpful comments. We also acknowledge useful discussions with Professor T. Nakajima. This work was partly supported by a Grant-in-Aid for Scientific Research (C) (23550004) and also in part by the Joint Usage/Research Program on Zero-Emission Energy Research, Institute of Advanced Energy, Kyoto University (B-22).

ε2(ωk)(2)(t ) − ε2(ωk −(2)1)(t ) = −λ 2(t )⟨ξ(k)(t )|{4α(θ ) ε2(ωk)(1)(t ) ) ) + β(θ ) εω(k(1) (t ) εω(k(2) (t )}|ψ (k)(t )⟩

tf

(k) whereby δf(k) 1 ≥ 0 is proved. Similarly, we can show δf 2 ≥ 0 (k,k−1) and therefore F ≥ 0, which proves the monotonic convergence of the iteration algorithm.

ε2(ωk)(1)(t ) − ε2(ωk −(1)1)(t )

+ β (θ )

∫0

− 1) ) + [εω(k(2) (t ) − εω(k(2) (t )]2 }

with the initial condition |ψ(k)(0)⟩ = |ψ0⟩. In eqs A5 and A6, the symmetrically divided electric fields associated with the 2ωcomponent are expressed as

) ) εω(k(1) (t ) εω(k(2) (t )}|ψ̅ (k)(t )⟩

(A12)

Substituting eqs A3 and A4 into eq A12 and then integrating both sides of eq A12 over t ∈ [0, tf], we obtain

1 ) ) α(θ)[εω(k(1) (t ) εω(k(2) (t ) + ε2(ωk)(1)(t ) ε2(ωk)(2)(t )] 4 ε2(ωk)(1)(t ) + ε2(ωk)(2)(t ) ⎫ ⎪ 1 ) ) ⎬|ψ (k)(t )⟩ − β(θ ) εω(k(1) (t ) εω(k(2) (t ) ⎪ 8 2 ⎭ 2

ε2(ωk −(1)1)(t ) + ε2(ωk −(2)1)(t ) ⎫ ⎪ ⎬ ⎪ 2 ⎭

) − 1) (t ) − εω(k(1) (t )]2 × |ψ (k − 1)(t )⟩[εω(k(1)

∂ iℏ |ψ (k)(t )⟩ ∂t

{BJ ̂

ε2(ωk −(1)1)(t ) + ε2(ωk −(2)1)(t ) ⎫ ⎪ ⎬ ⎪ 2 ⎭

− 1) (t ) + β(θ) εω(k(2)

with the final condition |ξ(k)(tf)⟩ = W|ψ̅ k)(tf)⟩, and

=

(k−1)

) − 1) (t ) − εω(k(2) (t )]2 × |ψ̅ (k)(t )⟩[εω(k(2) ⎧ ⎪ i (k) − 1) (t ) + ⟨ξ ̅ (t )|⎨2α(θ) εω(k(2) ⎪ 8ℏ ⎩

1 ) ) α(θ)[εω(k(1) (t ) εω(k(2) (t ) + ε2(ωk)(1)(t ) ε2(ωk −(2)1)(t )] 4 ε2(ωk)(1)(t ) + ε2(ωk −(2)1)(t ) ⎫ ⎪ 1 ) ) ⎬|ξ(k)(t )⟩ − β(θ ) εω(k(1) (t ) εω(k(2) (t ) ⎪ 8 2 ⎭

{BJ ̂

(k)

⎧ ⎪ d i (k) ) P1(t ) = (t ) ⟨ξ ̅ (t )|⎨2α(θ) εω(k(1) ⎪ dt 8ℏ ⎩

where the positive function, λ1(t), called a convergence parameter, characterizes the convergence behavior (speed, accuracy, etc.). Auxiliary step 2:

=

can be rewritten as

δf1(k) = δψ̅ (k , k − 1)(tf )|W |δψ̅ (k − 1)(tf ) + 2Re{P1(tf )}

(A4)

iℏ

δf(k) 1

(A8)

where the positive function, λ2(t), is another convergence parameter associated with the second auxiliary step. 11226



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Optimal Control Simulation of Field-Free Molecular Orientation

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