Let us now focus on the entanglement generated in our system. The cou-pled molecules can be expressed as a pure bipartite system. The reduced density operator for the first molecule is defined as
ρmol1 = Trmol2|Ψi hΨ| . (4.5)
To study the degree of entanglement, the bases of molecule 1 is transformed to make the reduced density matrix ρmol1 to be diagonal. The entangled state can be represented by a biorthogonal expression with positive real coefficients λlm
which can be obtained by diagonalization of density matrix ρmol1. The degree of entanglement for the coupled molecules can then be measured by von Neumann entropy [46, 47]
X
In our work, NaI molecule in the ground state with dipole moment 9.2 debyes and rotational constant 0.12 cm−1 is used. The field strength is 3 × 107 V/m and the laser frequency is about 9 × 1011 s−1. The duration and center of the pulse are set equal to 279 fs and 1200 fs. The main feature is that the ratio in magnitude of the positive and negative peak value of the laser pulse is 5 : 1. Unless specified, the parameters of the pulse are fixed throughout the chapter. The crossover from non-entangled case to entangled one is studied based on the initial condition:
c0,0;0,0(t = 0) = 1.
4.3 Results and discussion
After the coefficients cl1,m1;l2,m2(t) are determined, the orientations hcos θ1i and hcos θ2i can be evaluated immediately. Fig. 4.1 shows the orientations of the first and second molecules after a single laser pulse is applied on both molecules.
For R = 3 × 10−8 m, the behavior of the first molecule is quite close to that of a free rotor [20]. This is not surprising because the dipole interaction is weak for this molecule separation. However, as two molecules get close enough (Fig.
4.1(b)), both molecules orient disorderly, and the periodic behavior disappears.
This is because the dipole interaction is increased as the distance between the molecules is decreased, and the energy exchange between two molecules becomes more frequently. The regular orientation caused by the laser pulse is inhibited by the mutual interaction.
The populations of some low-energy levels are shown in the lower panels of Fig.
4.1(a) and (b). The solid, dashed, and dotted lines represent the populations of the states (1,0;0,0), (1,0;1,0), and (2,0;1,0), respectively. These states show different degrees of periodic behavior at different distances. However, the populations of some higher excited states, for example the (3,0;1,0) state in the inset of Fig.
4.1(b), display different degrees of irregularity. This manifests a fact that the nonlinear effect, caused by the reduction of R, does not affect the regularity of the low-lying states, and the origin of the irregularity is caused by the higher excited states.
Consider now the molecules are irradiated by a series of laser pulses periodi-cally. As shown in Fig. 4.2(a), if the period of the applied periodically laser pulse T is equal to ¯h/B , then both molecules behave disorderly no matter how the distance R is varied. The chaotic behavior of the molecules can be ascribed to the well-known ”kicked-rotor”problem. However, a series of regular-like orientations marked by dotted and dashed lines are present in Fig. 4.2(b) if T is equal to π¯h/B. For a free rotor under a single kick, this interesting phenomenon comes from the situation as the magnitude of the orientation returns to its initial con-dition (hcos θi = 0) after a certain period T [20]. Therefore, for two molecules in weak interaction limit (R = 3 × 10−8 m), the wavepacket-like orientation is similar to that of a single free rotor under the same laser period. The difference
the dipole force can generate some accidental phases to perturb the regularity of the coupled system. The lower panel of Fig 4.2(b) exhibits that the suppression of the regularity is quicker if the dipole force is stronger.
Fig. 4.3 shows the time-dependent entropy after one pulse passes through this system. For inter-distance R = 5 × 10−8 m, the entropy increases slowly from zero. For R = 1.5 × 10−8 m, on the contrary, the entropy grows rapidly with the increasing of time because the dipole force is stronger. Notes that the entropy only varies within a finite range at long time regime. This indicates that the systems reaches a dynamic equilibrium state even though the dipole force is still present.
Fig. 4.4(a) illustrates the variations of the entropy with respect to different field strengths of the applied laser pulse as R is set equal to 1.5 × 10−8 m. For the field strength E0 = 1.5× 107 V/m, an irregular-like behavior of the entropy is obtained, and its value is not large enough for quantum information processing.
However, Fig. 4.4(b) shows that the degree of entanglement can be enhanced if one increases the field strength. This can be understood well by studying the relationship between the dipolar interaction and the field strength. If the effect of dipole interaction overwhelms the laser field, most of the populations are distrib-uted on the low-lying states. In this case, the entropy from Schmidt decomposition is certainly small as shown in Fig. 4.4(a). On the other hand, if the field strength plays a dominant role, the distribution of molecular states covers a wider range
and the entropy is enhanced in this limit.
Next we detune the frequencies of the laser fields to study the behavior of entanglement. Figure 4.5 illustrates the time evolution of the entropy with dif-ferent ratios in magnitude of the positive and negative peak value of the laser pulse as R is set equal to 1.5 × 10−8 m. The laser frequency is tuned to change the ratio as shown in the inset by fixing other parameters. For the case of ratio 9 : 1, an irregular-like behavior is obtained with time-averaged value 0.51. If the ratio is set equal to 1 : 1, the entanglement shows a nearly periodic behavior with small averaged entropy. This result is very similar to the limiting case without laser, and indicates that the entanglement depends sensitively on the ratio of the laser pulse, i.e. the excitation is suppressed under the condition of 1 : 1 ratio.
Meanwhile, the dipole force only establishes periodic-like entropy.
Let us consider the first ten most contributive coefficients λl,m. The λl,m are re-arranged and denoted as λp with p = 1, 2, 3.... For example, λ1 is the most contributive coefficient. The insets of Fig. 4.5 show the ten coefficients (λp) at short and long time regimes. In the case of the ratio 9 : 1 , the eigenvalue λ1 dominates the contributions at short time regime(t = 50 ps). However, the contributions are distributed more averagely between different levels as t = 800 ps regime. This means the system is in some sort of dynamic equilibrium in long time limit, and entropy saturates to certain value. On the contrary, λ1 always
ratio 1 : 1 as shown in the lower inset of Fig. 4.5. From statistical point of view, this somehow explains the suppressed and regular behaviors of the entanglement (entropy).
Figure 4.6 shows the time evolutions of the populations of the eigenstates for different ratio of pulse shapes. For 1 : 1 ratio, the pulse hardly excites the rotors from the initial energy level (0, 0; 0, 0). Therefore, (0, 0; 0, 0) is still the mostly populated level (the population value is nearly close to 1 ) as shown in the lower panel of Fig. 4.6 while the pulse passes through. Similar to the ground state, the populations of the higher levels (the inset of Fig. 4.6) also show the periodic behavior. The periodic behavior is ascribed to the dipole interaction. Since the small fluctuation of the population is dominated by the dipole interaction in the case of symmetrical pulses. The magnitudes of the periodic fluctuations in higher level populations are rather small with the periodic evolution of the entropy. On the other hand, for 9 : 1 ratio the populations of the higher states show different degrees of irregularity as shown in the upper panel of Fig. 4.6. This is because a single asymmetrical pulse can generate high populations in the excited states [20], i.e. a larger angle orientation. The larger angle orientation can cause a largely fluctuated dipole interaction between the molecules. For this situation, energy transfer by means of (mediated) dipole interaction generates the irregular evolutions of the higher excited states which result in a randomly time-varying entropy.
We further study the entropy for different separation and dipole moment in Fig. 4.7. The ratio is set equal to 9 : 1. If the separation is smaller (0.8 R), the entropy grows faster. On the contrary, the entropy evolves slower for the case of larger separation. This means that the system needs much more time to approach the dynamic equilibrium. We also study the time evolution of entropy by changing the dipole moment. Our result shows that a similar behavior of the entropy exhibits, i.e. the strength of dipole interaction governs the behavior of evolution.
By adjusting the laser parameters, one can vary the degree of the entangle-ment. Figure 4.8 illustrates the time evolution of the entropy under single pulse or double pulses with ratio 5 : 1. As can be seen, an irregular behavior of the entropy is obtained, but their averaged values are different. For single kick, the populations are first dominated by this laser pulse. Then, the dipole interaction plays a key role to raise the entanglement in the system. In the case of double pulses the finite populations is created by first pulse. As the second laser pulse passes through, the populations will be redistributed to a wider range. Since the populations are distributed more averagely in this case, the entropy is certainly larger as shown by the solid line in Fig. 4.8. One can notes that the enhancement of entropy is achieved by applying the second laser pulse. Consider the case that time separation between these two pulses is set to be 5 times the center of the
tuned to obtain different degree of entropy. Another way to control the degree of entanglement in this system is to change the positive and negative ratios of the laser pulse. Inset of Fig. 4.8 shows the time-averaged entropy with respect to different ratios. We find that the entropy is more enhanced as the ratio is larger.
This means that the highly asymmetric laser pulse can generate larger entropy under the same field strength.
To study the crossover behavior from quantum to classical limit in this system, one can tune the fundamental Planck constant ¯h0. Figure 4.9 shows the time for entropy first exceeds the time-averaged value (arrow in the inset) versus the different factor of Planck constant ¯h0. As shown, the time grows rapidly with the decreasing of the Planck constant ¯h0. The inset in Fig. 4.9 shows a slowly increasing of entropy with the evolution of time for ¯h0 = 0.01¯h. Comparing this with the result for ¯h0 = ¯h, the ratio of the two times is roughly 100 : 1. This means that the entropy evolves slowly, and the system needs a longer time to approach dynamical equilibrium for a small ¯h0. As expected, the time for classical limit (¯h0 → 0) goes to infinity, satisfying that no entanglement exists between classical objects.
For a more realistic molecular system, one can extend our model to hindered-rotor system. The hindered hindered-rotor means that the polar diatomic molecule is adsorbed on the surface with the confinement of surface potential. In other words, one reasonably considers that two coupled polar molecules are adsorbed on the
surface with the dipole interaction. Comparing hindered rotor with free one, the rotation of a hindered rotor is similar to that for the free one, but the degree of orientation is different. This is because that the surface potential confines the rotation. Although this confinement may affect the property of the system, according to our work in chapter 3, a free rotor and hindered rotor actually show the same physics. In particular, a hindered rotor can be transformed into a free one by changing the parameters of surface potential.
0 1000 2000 3000
Figure 4.1: Upper panels of Fig. 4.1(a) and (b) show the orientations of the two molecules at different distances. Lower panels: The populations of the states (l1, m1; l2, m2)=(1, 0; 0, 0) (solid lines), (2, 0; 1, 0) (dotted lines), (1, 0; 1, 0) (dashed lines). The insets in (a) and (b) represent the population of state (3, 0; 1, 0) .
0 1000 2000 3000-0.8 0.0 -0.8 0.8
0.0
0.8 -0.8
0.0 0.8 -0.8
0.0 0.8
(b) T=πh/B
time [ps]
<cosθ>,<cosθ'>
(a) T=1h/B
Figure 4.2: The orientations of the first and second molecules under periodic laser pulses with the periods T= (a) 1¯h/B, (b) π¯h/B ps. The upper and lower panels of (a) and (b) correspond to the distances R = 3 × 10−8 and 2 × 10−8 m, respectively.
0 1000 2000 3000 4000 0.0
0.2 0.4 0.6
0 4000 8000 12000
0.0 0.2 0.4 0.6
(b) R=1.5x10-8 m
EntropyEntropy
time [ps]
(a) R=5x10-8 m
Figure 4.3: Time evolution of the entropy after applying single laser pulse for (a) R = 5 × 10−8 m and (b) R = 1.5 × 10−8 m.
0 1000 2000 3000 4000 0.0
0.2 0.4 0.6 0.8 0.0 0.1 0.2 0.3
(b) E
0=6x107 V/m
Entropy
time [ps]
(a) E
0=1.5x107 V/m
Figure 4.4: Time evolution of the entropy for inter-molecule separation R = 1.5
× 10−8 m. The degree of entanglement can be enhanced if one increases the field strength.
0 200 400 600 800 1000
Figure 4.5: Time evolution of the entropy after applying single laser pulse for different ratios in magnitudes of the positive and negative peak value of the laser pulse. The graphs show the irregular (periodic) behavior for ratio 9 : 1 (1 : 1).
The inset : the first ten contributive eigenvalues λp at short time (t = 50 ps) and long time (t = 800 ps).
0 200 400 600 800 1000 (dot-ted line). The inset in the lower panel is the enlarged figure showing the states (1, 0; 1, 0) (solid line), (1, 1; 1, 1) (dotted line), respectively.
0 200 400 600 800 1000
Figure 4.7: Time evolution of the entropy for different separation and dipole moment under single pulse (ratio 9 : 1). The dotted curve shows the case of R = 1.5× 10−8 m and µ = 9.2 D. The dashed and solid curves correspond to (a) 0.8 R and 1.2 R, or (b) 1.2 µ and 0.8 µ, respectively.
0 200 400 600 800 1000
Figure 4.8: Time evolution of the entropy for fixed ratio 5 : 1 under single pulse (dashed line) and double pulses (solid line). Time separation (tapp) between two pulses is set to be 5 times the center of the laser peak. The inset : Dependence of the time-averaged entropy on the pulse shape for inter-molecule separation R = 1.5 × 10−8 m.
0.0 0.2 0.4 0.6 0.8 1.0
0 10000 20000 30000 40000
0.0
Figure 4.9: The time with respect to different Planck constant ¯h0 for fixed field strength E0 = 3× 107 V/m and inter-molecule separation R = 1.5 × 10−8 m. The time is defined as the first time in entropy that exceeds the time-averaged value (the arrow in the inset). The inset : the time-averaged value (dotted line), and time evolution of the entropy for ¯h0 = 0.01¯h (solid line).
CHAPTER 5
COUPLED ADSORBED MOLECULES IN LASER FIELDS
In the complex surface systems, adsorbed molecules may not be isolated.
Several studies have shown that interesting behavior can occur due to the existence of dipole-dipole interaction [24, 25, 26, 27, 28]. In addition, since the investigations on entangled behavior of two coupled rotors are limited in the model of kicked tops [43, 44], this inspires us to study the dynamical entanglement of adsorbed molecules. According to our study in chapter 3, it is found that the orientations of free coupled rotors somehow reflect the entropy of the system and thus relate to the measurement of entanglement. Since the entanglement measurement is one of the fundamental important issues in quantum information research, the study of the entanglement and its measurement becomes an interesting problem. Moreover, from the experimental point of view, it is still not clear how to keep two free rotors with fixed distance. Therefore, this makes it more interesting to consider a more realistic system and discuss the corresponding entanglement dynamics.
In this chapter, we investigate the rotational motions of a polar diatomic
pulse, the hindered rotor shows periodic behavior. Different signatures between 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 si-multaneously ultra-short laser pulse shined upon them. It is found that both the entanglement (the von Neumann entropy) and orientation show interesting behaviors.
5.1 Single adsorbed molecule in a strong laser pulse
Consider now a dipolar molecule (e.g. NaI) adsorbed on the surface. The rotation of the molecule is confined by the surface potential as shown in Fig.
5.1. An off-resonant laser field polarized in z-direction interacts with the hindered rotor. Because the laser frequency is much lower than the frequencies of the lowest vibrational and electronic transition, only the rotational excitations can occur in our model. The excitations can be viewed as two photon transitions between two different rotational states through a high intermediate virtual state [18]. The Hamiltonian without the field-molecule interaction can be written as
H0 = BJ2+ Vhin(θ, φ), (5.1)
where B and J2 are rotational constant and angular momentum. Vhin denotes the surface potential and confines the rotation of adsorbed molecule. For simplicity,
the infinite-conical-well model Vhin(θ, φ) is considered here. According to the previous studies, its dependence on φ is weaker than that on θ [32, 33, 34]. We reasonably assume that the surface potential is independent of φ. Therefore, in the vertical adsorbed configuration, the surface potential can be written as [3]
Vhin(θ) =
where α is the hindering angle of the conical well.
The Hamiltonian concerning the field-molecule interaction can be written as
Hd =−µE (t) cos θ, (5.3)
Hind =−1
2E2(t) ((αk− α⊥) cos2θ + α⊥). (5.4) The first term Hddescribes a permanent dipole moment µ coupling with an exter-nal field, and θ is the angle between the molecular axis and the field. In this work we choose a Gaussian pulse for our calculation, i.e. E (t) = E0e−(t−t0)2/σ2cos (2πνt) , where E0 is the field strength and ν is the laser frequency. The pulse is centered at the time t0, and σ is the pulse duration. The second term Hind is a higher order interaction, in which the external field couples with the induced molecular polarization. The component of the polarizability αk (α⊥)is parallel (perpendicu-lar) to the molecular axis. According to our parameters, the field-dipole-moment interaction Hd is much greater than that of the field-induced-dipole-moment in-teraction Hind in our model. This is because the strength of electric field used
Hind can play an important role in the case of high strength of electric field [18].
Therefore, the term (Hind)can be neglected reasonably based on our parameters.
Before solving the time-dependent Schrödinger equation (H0+ Hd), the eigen-functions of the system (H0 = BJ2+ Vhin(θ)) must be introduced first. Following Ref. [3], the eigenfunctions can be written as
ψlm(θ, φ) =
where Al,m is the normalization constant and Pν|m|l,m is the associated Legendre Function of arbitrary order with the corresponding quantum number (l, m). In above equations, the molecular rotational energy can be expressed as
l,m = νl,m(νl,m+ 1)B. (5.6)
In order to determine νl,m, one has to match the boundary condition
Pν|m|l,m(cos α) = 0. (5.7)
To solve time-dependent Schrödinger equation, the wavefunction is expressed in terms of a series of eigenfunctions:
Ψ (t) = X
l
cl,m(t) ψl,m(θ, φ) , (5.8)
where cl,m(t)is time-dependent coefficient. The coefficient cl,m(t)can be obtained from the different equations
After determining the coefficients cl,m(t), the orientation hcos θi can be carried out immediately. We choose NaI as our model molecule, whose dipole moment µ = 9.2 Debye and rotational constant B =0.12 cm−1. For simplicity (zero-temperature case), the rotor is assumed in ground state initially, i.e. c0,0(t = 0) = 1. The field strength is 3×107 V/m and the laser frequency is about 9×1011s−1. The duration and center of the pulse are set equal to 279 fs 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.
Figure 5.2 illustrates the orientation hcos θi as a function of time for different hindering angles and pulse durations. In both cases, the orientations display periodic-like behavior. For the pulse duration (σ0 = σ), the orientation of small hindering angle (α = 60◦) shows a relative large value but with small oscillatory amplitude, while for α = 120◦ a large oscillatory amplitude with multi-frequency (insets of Fig. 5.2) is obtained. Obviously, such a difference comes from the quantum confinement effect. We further apply the laser pulses with different widths by tuning the duration and center. If the pulse duration increases, the amplitudes of the oscillations decrease 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
Figure 5.2 illustrates the orientation hcos θi as a function of time for different hindering angles and pulse durations. In both cases, the orientations display periodic-like behavior. For the pulse duration (σ0 = σ), the orientation of small hindering angle (α = 60◦) shows a relative large value but with small oscillatory amplitude, while for α = 120◦ a large oscillatory amplitude with multi-frequency (insets of Fig. 5.2) is obtained. Obviously, such a difference comes from the quantum confinement effect. We further apply the laser pulses with different widths by tuning the duration and center. If the pulse duration increases, the amplitudes of the oscillations decrease 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