Orientation of adsorbed dipolar molecules: A conical well model

Y. Y. Liao, Y. N. Chen, W. C. Chou, and D. S. Chuu*

Department of Electrophysics, National Chiao-Tung University, Hsinchu 300, Taiwan 共Received 14 July 2005; revised manuscript received 8 December 2005; published 21 March 2006兲

Orientation of single and two coupled polar molecules irradiated by a single laser pulse under a conical-well model is investigated theoretically. The orientation of a single hindered rotor shows a periodic behavior. In particular, the amplitude of the oscillation is sensitive to the degree of alternation of the laser field. Crossover from field-free to hindered rotation is observed by varying the hindering angle for different heights of conical wells. For a small hindering potential and angle, time-averaged orientation differs greatly from that for an infinite one. The orientation at a large hindering angle shows irregularlike behavior under strong dipole-dipole interaction. Entanglement induced by the dipole-dipole interaction is also calculated for the coupled-rotor system, in which the time-averaged entropy increases monotonically as the hindering angle is increased. The competition between the confinement effect and dipole interaction is found to dominate the behavior of the coupled-rotor system.

DOI:10.1103/PhysRevB.73.115421 PACS number共s兲: 73.20.Hb, 33.20.Sn, 33.80.⫺b

Controlling the orientation of molecules has wide appli-cations from stereodynamics to surface catalysis, molecular focusing, and nanoscale design.¹The molecular alignment is responsible for the anisotropic polarizability induced by the nonresonant laser pulses. For adiabatic regime, the crucial characteristic is that the duration of the laser pulse is longer than the rotational period. The pendular states can be created adiabatically, and the molecular axis is aligned parallel to the direction of the field polarization. As the laser pulse is switched off, the molecule will go back to its initial condi-tion and no longer be observed again.^2,3If the duration of the laser pulse is shorter than the rotational period, the alignment occurs periodically in time, i.e., the nonadiabatic regime.^4,5 On the other hand, a field-free orientation can be generated by a femtosecond laser pulse.^6,7 The dipole molecule will tend to orient by applying a highly asymmetrical pulse. It is found that a pronounced orientation can still persist after switching off of the pulse.

In addition to the rotation of a free rotor, the rotational motion of molecules adsorbed on a solid surface has at-tracted increasing interest. To understand the rotational be-havior of adsorbed molecules, one can apply the UV laser beam along the surface direction to desorb the molecules.

The rotational states can be determined by the quadrupole, which is a measure of the rotational alignment.^8–10 On the theoretical side, an infinite-conical-well model has been pro-posed. It was found that the adsorbed molecule is only al-lowed to rotate within the well region.¹¹In our earlier works, we have generalized the infinite-well model to a more real-istic case of finite-conical well.^12,13It was found that there exists the avoided crossing between two adjacent rotational energies when varying the strength of the external field. Our theoretical calculation of the quadrupole moment based on the finite-conical-well model is in agreement with the experi-mental data.¹⁴

Recently, great attention has been focused on the coupled-rotor system since peculiar behaviors may occur in the pres-ence of dipole-dipole interaction.^15–17 For example, recent neutron scattering experiments on certain Hofmann clath-rates have reported the temperature-dependent behavior of

the linewidths.¹⁸ A line broadening mechanism based on rotor-rotor coupling was proposed for the explanation of the widths.¹⁹With the advances of nanotechnology, one can now investigate the quantum rotors which are mounted on the surfaces.^20,21 From the laser spectroscopy, two individual fluorescent molecules separated by several nanometers can be resolved on the surface.²² The coherent interactions be-tween the dipole moments associated with their optical tran-sitions are found in the quantum optical measurements. The strong dipole-dipole coupling produces entangled subradiant and superradiant states in the two-molecule-system under la-ser radiation.

Even though the orientation of the free molecule is well studied, investigations on the rotation of the adsorbed mol-ecule confined the surface potential are still lacking.²³ In the complex surface system, the adsorbed molecules are no longer isolated. Several studies have shown that interesting phenomena can occur due to the existence of dipole-dipole interaction. Besides, although the entangled behavior of two coupled rotors was also investigated recently, these works are limited in the model of kicked tops.^24,25 However, the dynamical entanglement via the rotations of the adsorbed molecules remains mostly unexplored. In particular, a mo-lecular system evolves from a nonentangled case to an en-tangled one. According to our previous study,¹⁷ it is found that the orientations of the coupled rotors relate closely to the entropy. This means that the orientations of the coupled ro-tors somehow reflect the entropy of the system and thus re-lates to the measurement of the entanglement. Since the mea-surement of the entanglement is one of the fundamental important issues in quantum information research, therefore, the study of the entanglement and its measurement is one of the interested problems. Moreover, from the experimental point of view, it is not clear how to keep two free rotors with a fixed distance. Therefore, this makes it more interesting to consider a more realistic system and discuss the correspond-ing entanglement dynamics.

In this paper, we first investigate the rotational motions of a polar diatomic molecule confined by a hindering conical well. After applying a single strong laser pulse, the hindered

1098-0121/2006/73共11兲/115421共5兲/$23.00 115421-1 ©2006 The American Physical Society

rotor shows a periodic behavior. Different signatures be-tween the finite-conical-well and infinite-conical-well model on orientations are pointed out. Besides, the amplitudes of the oscillations are varied by applying different widths of the pulse. Furthermore, we also consider two coupled identical polar molecules adsorbed on the surface with the dipole-dipole interaction and a simultaneously ultrashort laser pulse shined upon them. It is found that both the entanglement共the von Neumann entropy兲 and orientation show interesting be-haviors.

Consider now a dipolar molecule共e.g., NaI兲 adsorbed on the surface. The rotation of the molecule is confined by the surface potential as shown in Fig. 1. An off-resonant laser field polarized in the z direction interacts with the hindered rotor. Because the laser frequency is much lower than the frequencies of the lowest vibrational and electronic transi-tion, only the rotational excitations can occur in our model.

The excitations can be viewed as two photon transitions be-tween two different rotational states through a high interme-diate virtual state.⁵ The Hamiltonian without the field-molecule interaction can be written as

H₀= BJ²+ Vhin共␪,␾兲, 共1兲 where B and J² are the rotational constant and angular mo-mentum. Vhin denotes the surface potential and confines the rotation of adsorbed molecule. For simplicity, the infinite-conical-well model Vhin共␪^,␾兲 is considered here. According to the previous studies, its dependence on␾is weaker than that on␪^.26–28We reasonably assume that the surface poten-tial is independent of␾. Therefore, in the vertical adsorbed configuration, the surface potential can be written as¹¹

V_hin共␪兲 =

再

where␣ is the hindering angle of the conical well.

The Hamiltonian concerning the field-molecule interac-tion can be written as

Hd= −␮E共t兲cos␪, 共3兲

H_ind= −1

2E²共t兲关共␣储−␣_⬜兲cos²␪⁺␣_⬜兴. 共4兲 The first term Hd describes a permanent dipole moment

␮ coupling with an external field, and ␪ is the angle

between the molecular axis and the field. In this work we choose a Gaussian pulse for our calculation, i.e., E共t兲

= E₀e^{−共t − t0兲2/␴2}cos共2␲␯^t兲, where E0is the field strength and␯ is the laser frequency. The pulse is centered at the time t₀, and␴is the pulse duration. The second term Hindis a higher order interaction, in which the external field couples with the induced molecular polarization. The component of the polar-izability ␣储共␣_⬜兲 is parallel 共perpendicular兲 to the molecular axis. According to our parameters, the field-dipole-moment interaction Hdis much greater than that of the field-induced-dipole-moment interaction Hindin our model. This is because the strength of electric field used here is unsufficient to en-hance the higher order term. Actually the interaction Hindcan play an important role in the case of high strength of electric field.⁵Therefore, the term共Hind兲 can be neglected reasonably based on our parameters.

Before solving the time-dependent Schrödinger equation 共H0+ Hd兲, the eigenfunctions of the system 关H0= BJ²+ Vhin共␪兲兴 must be introduced first. Following Ref.

11, the eigenfunctions can be written as

␺lm共␪^,␾兲 =

冦

^冑

where Almis the normalization constant and P_␯

lm

兩m兩 is the asso-ciated Legendre function of arbitrary order with the corre-sponding quantum number共l,m兲. In the above equations, the molecular rotational energy can be expressed as

⑀lm=␯lm共␯lm+ 1兲B. 共6兲 In order to determine ␯lm, one has to match the boundary condition

P_␯

lm

兩m兩共cos␣兲 = 0. 共7兲

To solve the time-dependent Schrödinger equation, the wave function is expressed in terms of a series of eigenfunc-tions

⌿共t兲 =

兺

where clm共t兲 is the time-dependent coefficient. The coeffi-cient c_lm共t兲 can be obtained from the different equations

iបc˙lm共t兲 = clm共t兲⑀lm+

兺

l⬘

cl⬘^m共t兲具␺lm兩Hd兩␺l⬘^m典. 共9兲

After determining the coefficients clm共t兲, the orientation 具cos␪典 can be carried out immediately. We choose NaI as our model molecule, whose dipole moment ␮= 9.2 D and rota-tional constant B = 0.12 cm⁻¹. For simplicity 共zero-temperature case兲, the rotor is assumed in the ground state initially, i.e., c₀₀共t=0兲=1. The field strength is 3⫻10⁷V / m and the laser frequency is about 9⫻10¹¹s^-1. The duration and center of the pulse are set equal to 279 and 1200 fs. The main feature is that the ratio in magnitude of the positive and negative peak value of this pulse is 5:1. Unless specified, the parameters of laser field are fixed throughout the paper.

FIG. 1.共a兲 Schematic view of a single hindered rotor adsorbed on the surface.共b兲 The corresponding infinite-conical-well model.

Figure 2 illustrates the orientation具cos␪典 as a function of time for different hindering angles and pulse durations. In both cases, the orientations display periodiclike behavior. For the pulse duration共␴⬘⁼␴兲, the orientation of small hindering angle 共␣= 60°兲 shows a relative large value but with small oscillatory amplitude, while for ␣= 120° a large oscillatory amplitude with multifrequency共insets of Fig. 2兲 is obtained.

Obviously, such a difference comes from the quantum con-finement effect. We further apply the laser pulses with dif-ferent widths by tuning the duration and center. If the pulse duration increases, the amplitudes of the oscillations de-crease and the orientations approach the initial value as shown in the insets. The reason is that the mean orientation is suppressed by the alternations of the electromagnetic field, i.e., the cancellation of negative and positive orientations.

To see more clearly the effect of the hindering potential, let us now consider the finite potential model

Vhin共␪兲 =

再

where V₀ is the height of well. Following Refs. 12–14, the rotational energy and eigenfunctions can be determined by matching appropriate boundary condition. Figure 3 shows the time-averaged orientation as a function of time for dif-ferent hindering potentials. For infinite potential共V0=⬁兲, the time-averaged orientation decreases monotonically from 1 to 0 as the hindering angle is increased. However, if the well is finite, the time-averaged orientation has a maximum point at certain angle. This means if the open angle␣decreases

fur-ther, the contribution from the penetrated wave function overwhelms the impenetrable one, rendering the decreasing of the time-averaged orientation. We also compare the case of␴⬘^=␴with that of␴⬘^{= 5␴}共inset of Fig. 3兲. It is found that, for larger duration ␴⬘^{= 5}␴, although the oscillatory ampli-tude is smaller共Fig. 2兲, the value of time-averaged orienta-tion is larger comparing to the case of␴⬘⁼␴^.

As we mentioned above, the spatial resolution of two in-dividual molecules hindered on a surface in tens of nanom-eters is now possible.^20–22We further consider that two iden-tical dipolar molecules共separated by a distance of R, R is in an order of magnitude of 10⁻⁸m兲 confined by the hindering wells. The molecules are assumed to interact with each other via dipole-dipole interaction only. A polarized laser pulse is applied to interact with both molecules. The Hamiltonian of the coupled system can be written as

Hc=

兺

j=1,2

H_0,j+ Udip+ HI, 共11兲 where H_0,jis the Hamiltonian of single hindered rotor with-out the laser-dipole interaction. The dipole interaction be-tween two dipole moments␮1 and␮2is molecule, respectively. For simplicity, we assume the dipole moments of two molecules are identical, i.e., ␮1=␮2=␮^. One might argue that the higher order terms may also con-tribute to the results. According to previous study,¹⁵the next higher order term is about the order of r³/ R⁴ with bond length r. If one compares the dipole-dipole interaction, 关O共r²/ R³兲兴, with the next higher order effect 关the bond length r = 2.7 Å共Ref. 29兲 and separation R=15 nm兴, it is found that the contribution from the next higher-order term is only 2%

FIG. 2. 共Color online兲 The orientation 具cos␪典 as a function of time for different hindering angle␣ and pulse duration ␴⬘^{. The} insets show the corresponding populations of the states共l,m=0兲 for 共a兲␣=60° and 共b兲 ␣=120°, respectively. The corresponding laser fields are shown in the upper inset.

FIG. 3. 共Color online兲 The mean orientation 具cos␪典mean as a function of hindering angle for fixed pulse duration 共␴⬘^{=␴兲 and} different conical-well potentials V₀= 10, 30, 100. The inset shows the mean orientation具cos␪典mean in the case of V₀= 10 and ⬁ by applying a pulse of ␴⬘^{= 5}␴. The potential V0 is in units of the rotational constant B.

of the dipole-dipole interaction. Therefore, it is reasonable to include only the dipole interaction in our model. The field-molecule coupling HI can then be expressed as

H_I= −␮^E共t兲cos␪1cos共␻^t兲 −␮^E共t兲cos␪2cos共␻^t兲, 共13兲 where␪1and␪2are the angles between dipole moments and laser field. In the above equations, the time-dependent Schrödinger equation can be solved by expanding the wave function in terms of a series of eigenfunctions

⌿c=_l

兺

1m₁l₂m₂

c_l

1m₁l₂m₂共t兲␺l₁m₁共␪1,␾1兲␺l₂m₂共␪2,␾2兲, 共14兲 where共␪1,␾1兲 and 共␪2,␾2兲 are the coordinates for two mol-ecules. cl₁m₁l₂m₂共t兲 are the time-dependent coefficients corre-sponding to the quantum numbers共l1, m₁; l₂, m₂兲, and can be determined by solving Schrödinger equations numerically.

The initial state is set as␺00␺00关c0000共t=0兲=1兴.

In addition to the orientation, one can also analyze the entanglement induced by the dipole interaction. The wave function of the coupled molecules can be expressed as a pure bipartite system 关a compact form of Eq. 共14兲兴:

兩⌿c典=兺l₁m₁l₂m₂cl₁m₁l₂m₂共t兲兩␺l₁m₁典兩␺l₂m₂典. The reduced density operator for the first molecule is defined as

␳mol 1= Tr_{mol 2}兩⌿c典具⌿c兩. 共15兲 To obtain the entanglement of entropy, the bases of molecule 1 is transformed to make the reduced density matrix␳mol 1to be diagonal. The entangled state can be represented by a biorthogonal expression with positive real coefficients ␭lm. The degree of entanglement for the coupled molecules can be measured by von Neumann entropy^30,31

Entropy = −

兺

Figure 4 shows the entropy and orientation evolves with time for fixed angle ␣= 120° and interdistance R = 1.5⫻10⁻⁸m. Because of the presence of the laser pulse, contributions to the energy exchange between two molecules come from many excited states, resulting in an irregularlike behavior of the entropy shown in Fig. 4共a兲. Further analysis of the dynamics gives the fact that the entropy grows mono-tonically from zero to a certain finite value. This is because the laser pulse dominates at the initial stage. The strength of the laser pulse is much larger than that of the dipole-dipole interaction. In addition, the duration is much shorter than the characteristic time of the dipole interaction. After the laser pulse, populations to the 共rotational兲 excited states are formed共inset兲. The nonlinear dipole interaction then initiates the exchange process between the states until certain “dy-namical equilibrium” is reached. One can conclude that the nonlinear variations of populations confirm the feature shown in the inset. Moreover, the orientations of the coupled molecules are also displayed in Fig. 4共b兲. Compared to the single molecule case, the irregular behavior is certainly from the nonlinear dipole interaction.

Figure 5 shows the time-averaged entropy for different hindering angles. As the hindering angle increases, the

time-averaged entropy increases monotonically. This is because for larger angles more excited states can be obtained under the same strength of the laser pulse, resulting in larger en-tropy. Note that the magnitude of orientation is high as the hindering angle is set equal to 30° 共inset of Fig. 5兲. This again verifies that the narrow potential restricts the motion of the hindered rotor. In this case, the dipole interaction is sup-pressed, causing the regularlike behavior of the orientation.

On the contrary, more excitations are populated such that the orientation oscillates with irregularity at␣^{= 150°.}

A few remarks about the experimental verifications of our model should be addressed here. According to our previous FIG. 4. 共Color online兲 The entropy 共a兲 and orientations 具cos␪1典共具cos␪2典兲 共b兲 in an infinite conical well for a fixed angle

␣=120° and interdistance R=1.5⫻10⁻⁸m. The inset shows the populations 共兩cl₁m₁l₂m₂共t兲兩²兲 irregularly oscillate with time, corre-sponding to the quantum number共l1, m₁; l₂, m₂兲=共1,0;0,0兲 共black solid curve兲, 共1,1;0,0兲 共red dashed curve兲, and 共1,0;1,0兲 共green dot-ted curve兲, respectively. Although we only focus on several excited states here, the populations of most states similarly remain irregular behavior.

FIG. 5. The time-averaged entropy as a function of the hindered angle in the infinite conical well. The insets show the orientations of two molecules for hindered angles ␣=30° and ␣=150°,

FIG. 5. The time-averaged entropy as a function of the hindered angle in the infinite conical well. The insets show the orientations of two molecules for hindered angles ␣=30° and ␣=150°,