The solid lines in the insets of Fig. 3.3 show the dependence of the alignment on hindered angle α. For α = 60◦, sinusoidal-like behavior is presented, and the alignment ranges from 0.63 to 0.91. As the hindered angle increases, the curves become more and more complicated and gradually approach the free rotor limit
can be understood well by studying the populations |cl,m|2 of low-lying states. In the regime of small hindered angle, there is little chance for electron to populate in higher excited states since the shrinking of the conical-well angle causes the increasing of energy spacings.
One also notes that the populations of a hindered molecule for α = 60◦ and 120◦, shown in Fig. 3.4(a) and (b), mainly compose of l = 0, 1 and 2 states, while the population of a free rotor is composed of l = 0, 2, 4 states. The underlying physics comes from the reason that
ψl0,m0|cos2θ| ψl,m
® is non-zero for all l and
l0 values in the case of hindered rotation. But it is zero in free rotor limit except for l = l0 or l = l0± 2. The dotted lines in the insets represent the first two main contributions of the factors P
l6=l0
ψl0,m0|cos2θ| ψl,m
®summed from low-lying states,
i.e. the sum of the largest two values of the off-diagonal term
ψl0,m0|cos2θ| ψl,m
®.
As can be seen, the populations for small hindered angle are mainly distributed on lower states since the main oscillation feature (e.g. the frequency) of the curve (dotted lines) is quite similar to that from whole contributions (solid lines).
Let us now turn our attention to the case of orientation. After applying a short pulse laser, the orientation hcos θi of a hindered molecule (α = 60◦) oscil-lates sinusoidally with time as shown in Fig. 3.4(a). The value of hcos θi is always positive because the rotational wavefunction is compressed heavily. As the hin-dered angle α becomes larger, the oscillation frequency also decreases as shown in Fig. 3.4(b). These signatures are quite close to that of the alignment. We
then conclude that even at larger hindered angle (α = 1200) the role of hindered potential still overwhelms the laser pulse, otherwise, the value of hcos θi should not always be positive.
Fig. 3.4(c) represents results of orientations in infinite (V0 =∞) or finite (V0 = 100) conical-well potential for α = 1750. Dashed and dotted lines correspond to V0 = ∞ and 100, respectively. For the case of finite conical-well potential, the wavefunction is expressed in terms of a series of the basis wavefunctions obtained in Refs. [4, 5, 7]. As can be seen, the effect of laser pulse is obvious because negative value appears. Comparing the results with the free orientation [20], the angular distributions for finite well are more isotropic since the wave functions can penetrate into the conical-barrier.
Further analysis shows that components of orientation hcos θi or alignment hcos2θi can be divided into two parts: diagonal and nondiagonal terms. The nondiagonal term represents the variations of these curves such as those in the insets of Fig. 3.3. These variations with time are determined by the phase differ-ence coming from various energy levels. To see the contributions from diagonal terms, we evaluate the time-averaged orientation and alignment. In this case, the nondiagonal values will be averaged out, and only contributions from diagonal terms exit. Fig. 3.5 shows the mean orientation and alignment as a function of hindered angle. As α increases, the mean orientation decreases monotonically
ψl,m|cos θ| ψl,m
®. For a larger angle α, the populations |cl,m|2 mainly compose
of l = 0, 2, 4 states. But the value
ψl,m|cos θ| ψl,m
® is governed by the selection
rule: l = l0 + 1. Thus the net effect is the shrinking of the mean orientation in large angle limit.
Contrary to orientation, the mean alignment shows a quite different feature.
The value of hcos2θi first decreases as α increases. However, it reaches a minimum point about for α = 1400. From the insets of Fig. 3.5, we know that the values of ¡
ψl,m|cos2θ| ψl,m
®¢ do not depend significantly on α. Therefore, the decrease
of hcos2θi comes from the decreasing tendency of the population |cl=1,m|2, while its increasing behavior is caused by other two populations |cl=0,m|2 and |cl=2,m|2. Competition between these two effects results in a minimum point.
Figure 3.1: Schematic view of the hindered rotor.
Figure 3.2: Quadrupole moments for the desorption of CO from Cr2O3(0001) as function of quantum number J. Filled circles: experimental data points.
0 1 2 3 4 5 6
Figure 3.3: The populations of the states (l, m = 0) for different hindered angles:
(a) α = 600, (b) α = 1200, (c) α = 1800. The insets show the correspond-ing alignments (solid lines) and the first two main contributions of the factors
P
l6=l0
ψl0,m0|cos2θ| ψl,m
®(dotted lines).
0 100 200 300 400 500 -0.5
0.0 0.5 0.0 0.5 1.0
0 50 100 150
0.8 0.9 1.0
(c)
time [ps]
(b)
<cosθ><cosθ><cosθ>
(a)
Figure 3.4: The orientations hcos θi (solid lines) of a hindered molecule confined by infinite conical-well for different hindered angles: (a) α = 600, (b) α = 1200, (c) α = 1750. The dashed and dotted lines in (c) correspond to different potential barrier height, i.e. V0 =∞ and 100, respectively.
30 60 90 120 150 180
Figure 3.5: The mean orientation hcos θimean and alignment hcos2θimean in infi-nite conical-well. The insets show the populations |cl,m|2 (fulled bar) and factors
ψl,m|cos2θ| ψl,m
® (sparse bar). Insets (a) and (b) correspond to α = 1100 and
α = 1700,respectively.
CHAPTER 4
COUPLED FREE MOLECULES IN LASER FIELDS
Recently, coupled-rotor- model attracts much interest because some physi-cal properties such as dielectric response may display peculiar behaviors in the presence of dipole-dipole interaction. In some materials, molecules are found to show a free rotation. For example, NH3 groups behave like one-dimensional quantum rotors in certain Hofmann clathrates [25]. In particular, a line broad-ening mechanism is proposed based on rotor-rotor coupling. With the advances of nanotechnology, one can investigate the quantum rotors which are mounted on the surfaces [29, 30, 31]. From the laser spectroscopy, two individual fluo-rescent molecules separated by several nanometers on the surface of an organic crystal can be resolved. The coherent interactions between the dipole moments associated with their optical transitions are found in the quantum optical mea-surements. The strong dipole-dipole coupling produces entangled subradiant and superradiant states in the two molecules system under laser radiation [30].
Many efforts have been devoted to generate entanglement in quantum-optic and atomic systems. Although some studies have been investigated on quantum rotors, these works are limited in the model of kicked tops [43, 44]. In this chapter, we consider a more realistic system. A method is proposed to create entanglement
between two coupled identical polar molecules separated in a distance of tens of nanometers. Both molecules are assumed to be irradiated simultaneously by the laser pulses. It is found that the entanglement induced by the dipole interaction can be affected by controlling the inter-molecule distance, the field strength, and the number of laser pulses. Moreover, the crossover from quantum to classical limit is also discussed by varying the Planck constant.
4.1 Model of two coupled free molecules in a strong laser pulse
Consider now two diatomic polar molecules (e.g. NaI) separated in a dis-tance of R. The molecule system is irradiated by half-cycle pulses. The total Hamiltonian can be written as
H = X
j=1,2
¯ h2
2IL2j + Udip+ HI, (4.1) where L2j and ¯h2I2 (= B) are the angular momentum operator and rotational con-stant, respectively. Udip is the dipole interaction between two molecules:
Udip = [µ1· µ2− 3 (µ1· beR) (µ2· beR)]
R3 , (4.2)
where µ1 and µ2 are the dipole moments. The dipole moments of two molecules are assumed, for simplicity, to be identical, i.e. µ1 = µ2 = µ0. The field-molecule coupling HI can thus be expressed as
X
where θ1 and θ2 are angles between dipole moments and laser field. The laser field is given by E (t, v) = E0f (t) cos (2πvt) ,where E0 is the field strength and v is the frequency. The envelope function f (t) is assumed to be Gaussian shape centered at the time t = t0 with duration σ, i.e. f (t) = e−(t−t0)2/σ2. Traditionally, a half-cycle pulse is a strongly asymmetric monohalf-cycle pulse that consists of two parts: a very short, strong pulse and a much long and weak tail of opposite electrical field.
The pulses E (t, v) used in the present work are actually not the exact half-cycle pulses as defined in Ref. [45]. However, practical calculation shows that there is almost no influence on our final result if a long and weak tail is introduced in the pulses E (t, v) = E0f (t) cos (2πvt). Thus, it is reasonable to model a half-cycle pulse by using the function E (t, v) in our calculation. In addition, the field duration is considered to be much shorter than the molecular rotational period in our work. Based on these conditions, an impulsive model can be employed in this case [20, 21]. The time-dependent Schrödinger equation can be solved by expanding the wave function Ψ in terms of a series of field-free spherical harmonic functions Yl,m(θ, φ)as
|Ψi = X
l1,m1;l2,m2
cl1,m1;l2,m2(t)|Yl1,m1(θ1, φ1)i |Yl2,m2(θ2, φ2)i , (4.4)
where (θ1, φ1) and (θ2, φ2)are the coordinates of the first and second molecule re-spectively. The time-dependent coefficients cl1,m1;l2,m2(t) correspond to the quan-tum numbers (l1, m1; l2, m2)and can be determined by solving Schrödinger equa-tions numerically. In equation (4.4), the inter-molecule separation R is assumed to
be fixed for simplicity, so that the total wavefunction has no spatial dependence.
Although the variation of R might be inevitable due to the influence of laser fields or inter-molecule vibrations, however, recent experiments exhibited that the spa-tial resolution in tens of nanometers for two individual molecules hindered on a surface is practically possible [29, 30, 31]. In principle, the free orientation model can be easily generalized to the hindered ones by replacing the spherical harmonic functions with hindered wavefunctions.