Antisymmetric states superposition

While the above process is of great scientific interest, practically speaking we usually want to separate a racemic mixture of the B and B enantiomers, that is, our typical initial state is a racemate. If we were to use the scenario of Section 8.2 to accomplish this separation one would have first to prepare the BAB adduct in a pure state. Since the preparation of fire BAB adduct, and especially its separation from the BAB and B AB adducts that would inevitably accompany its preparation, is not a trivial task, it is preferable to find control methods that could separate the B and B racemic mixture directly. In this section we outline a method that can achieve this much more ambitious task. The essential principles of this method remain the same as in Section 8.2, that is, excitation of a superposition of symmetric and antisymmetric states with respect to ah, the reflection operation. 

A number of interesting conclusions follow from Eq. (81). In the first place, we note that the superposition states decay at different rates, the symmetric state decays with an enhanced rate (F I T ), while the antisymmetric state decays at a reduced rate (r — 1)2). For F12 = T, the antisymmetric state does not decay at all. In this case the antisymmetric state can be regarded as a dark state in the sense that the state is decoupled from the environment. Second, we note from Eq. (81) that the state a) is coupled to the state j) through the splitting A, which plays a role here similar to the Rabi frequency of the coherent interaction between the symmetric and antisymmetric states. Consequently, an initial population in the state a) can be coherently transferred to the state j), which rapidly decays to the ground state. When A = 0, that is, the excited states are degenerate, the coherent interaction does not take place and then any initial population in a) will stay in this state for all times. In this case we can say that the population is trapped in the state u). 

The equation of motion (115) allows us to analyze conditions for population trapping in the driven A system. In the steady state (p = 0) with p / 1 and Ac = 0 the population in the upper state p33 = 0. Thus the state 3) is not populated even though it is continuously driven by the laser. In this case the population is entirely trapped in the antisymmetric superposition of the ground states. This is the CPT effect. However, for p 1 and Ac = 0, the antisymmetric state decouples from the interactions, and then the steady-state population p33 is different from zero [46]. This shows that coherent population trapping is possible... 

Thus, the condition V 2 = /1 1I2 for suppression of spontaneous emission from the antisymmetric state is valid for identical as well as nonidentical atoms, whereas the coherent interaction between the superpositions appears only for nonidentical atoms with different transition frequencies and/or spontaneous damping rates. 

In the section IV.B.2, we have shown that two nonidentical two-level atoms can be prepared in an arbitrary superposition of the maximally entangled antisymmetric state a) and the ground state g)... 

Similar to the fluorescence intensity distribution, the visibility can provide us an information about the internal state of the system. When the system is prepared in the antisymmetric state or in a superposition of the antisymmetric and the ground states, p55 = pee = 0, and then the visibility has the optimum negative value V = — 1. On the other hand, when the system is prepared in the symmetric state or in a linear superposition of the symmetric and ground states, the visibility has the maximum positive value "V - 1. 

The presence of two minima always leads to a modification of the harmonic vibrational structure. If the two minima are equivalent, the lowest level will have one symmetric and one antisymmetric proton wave function. The larger the barrier, the closer the two lowest vibrational levels, until they become almost degenerate. The wave function of the proton is then a superposition of the symmetric state and the antisymmetric state, localized in either of the two minima. 

Fig. 8.2 Schematic showing controlled dissociation of molecule B-A-B to yield B-A+B, or B + A-B products, where B and B are two enantiomers. A molecule is excited from ah 3 initial state i i) to a superposition of antisymmetric (] 2)) and symmetric (]/ 3)) vibratiorialf states belonging to an excited electronic state, by excitation pulse x(a>). After an appropriate delay time, the molecule is dissociated by second pulse ed(ca), to the E, n, D ) or E, n, continuum state. fv I ...

The procedure that we propose to enhance the concentration of a particulap enantiomer when starting with a racemic mixture, that is, to purify the mixture) is as follows [259], The mixture of statistical (racemic) mixture of L and irradiated with a specific sequence of three coherent laser pulses, as described below. These pulses excite a coherent superposition of symmetric and antisymmetric vibrational states of G. After each pulse the excited system is allowed to relax bg t to the ground electronic state by spontaneous emission or by any other nonradiativ process. By allowing the system to go through many irradiation and relaxatio cycles, we show below that the concentration of the selected enantiomer L or can be enhanced, depending on tire laser characteristics. We call this scenario lat distillation of chiral enantiomers. 

The results in this chapter make clear that a chiral outcome, the enhancement off j particular enantiomer, can arise by coherently encoding quantum interference infqjS mation in the laser excitation of a racemic mixture. The fact that the initial stall displays a broken symmetry and that the excited state has states that are eith jj symmetric or antisymmetric with respect to ah allows for the creation of a si position state that does not have these symmetry properties. Radiatively couplingfhf states in the superposition then allows for the transition probabilities from L and fi t differ, allowing for depletion of the desired enantiomer. 

Thus the excitation pulse can create a superposition of i), 2) consisting of two states of different reflection symmetry. The resultant superposition possesses no symmetry properties with respect to reflection [78]. We now show that the broken symmetry created by this excitation of nondegenerate bound states translates into a nonsymmetry in the probability of populating the degenerate , n, D ), , n, L ) continuum states upon subsequent excitation. To do so we examine the properties of the bound-free transition matrix elements ( , n, q de,g Ek) that enter into the probability of dissociation. Note first that although the continuum states , n, q ) are nonsymmetric with respect to reflection, we can define symmetric and antisymmetric continuum eigenfunctions , n, s ) and , n, a ) via the relations... 

The two intragap states y/ x) are the symmetric and antisymmetric superpositions of the midgap states localized near the soliton and the antisoliton ... 

An alternative way of viewing the process of the reduction of the dressed trapping state to the state a) in a very strong field is to analyze the equations of motion for the density matrix elements in terms of the symmetric and antisymmetric superpositions of the atomic excited states... 

Since the driven and undriven transitions are coupled through the Tn terms, it is convenient to introduce symmetric and antisymmetric superposition states of the dressed states +, V) and 3, V). According to Eq. (68), the superposition states diagonalise the dissipative (damping) part of the master equation of the system. The superposition states can be written as [51]... 

As we have shown in Sec. III. A, the second-order correlation function of the fluorescence field depends on correlation functions of the atomic dipole moments (S+(f)S+(f + x)Sy(t)Sj (t)), which correspond to different processes including photon emissions from a superposition of the excited levels. Therefore, we write the correlation functions G (R, t) and G (R, t R, t + x) in terms of the symmetric and antisymmetric superposition states as... 

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