Superposition states quantum interference

Superposition States and Interference Effects in Quantum Optics... 

The essential principle of coherent control in the continuum is to create a linear superposition of degenerate continuum eigenstates out of which the desired process (e.g., dissociation) occurs. If one can alter the coefficients a of the superposition at will, then the probabilities of processes, which derive from squares of amplitudes, will display an interference term whose magnitude depends upon the a,. Thus, varying the coefficients a, allows control over the product properties via quantum interference. This strategy forms the basis for coherent control scenarios in which multiple optical excitation routes are used to dissociate a molecule. It is important to emphasize that interference effects relevant for control over product distributions arise only from energetically degenerate states [7], a feature that is central to the discussion below. 

We end this section with a comparison of the basic concepts of laser control and traditional temperature control. This discussion includes an elementary explanation and definition of concepts such as incoherent superpositions of stationary states versus coherent superpositions of stationary states and quantum interference. 

Control of the type discussed above, in which quantum interference effects are used to constructively or destructively alter product properties, is called coherent control (CC). Photodissociation of a superposition state, the scenario described above, will be seen to be just one particular implementation of a general principle of coherent control Coherently driving a state with phase coherence through multiple, coherent,... 

This scenario opens up a wide range of possible experimental studies of control bimolecular collisions. Specifically, we need only prepare A and A in a control superposition of two states [e.g., by resonant laser excitation of A(1))] to produf superposition with r/>A(2)), direct them antiparallel in the laboratory, and vary t coefficients in the superposition to affect the reaction probabilities. Control -originates in quantum interference between two degenerate states associated wiili, r the contributions of 0A(1)) A<(2)> and I[Pg.154]

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. 

The effect of quantum interference on spontaneous emission in atomic and molecular systems is the generation of superposition states that can be manipulated, to reduce the interaction with the environment, by adjusting the polarizations of the transition dipole moments, or the amplitudes and phases of the external driving fields. With a suitable choice of parameters, the superposition states can decay with controlled and significantly reduced rates. This modification can lead to subnatural linewidths in the fluorescence and absorption spectra [5,10]. Furthermore, as will be shown in this review, the superposition states can even be decoupled from the environment and the population can be trapped in these states without decaying to the lower levels. These states, known as dark or trapped states, were predicted in many configurations of multilevel systems [11], as well as in multiatom systems [12],... 

The CPT effect and its dependence on quantum interference can be easily explained by examining the population dynamics in terms of the superposition states. v) and a). Assume that a three-level A-type atom is composed of a single upper state 3) and two ground states 1) and 2). The upper state is connected to the lower states by transition dipole moments p31 and p32. After introducing superposition operators 5+ = (S ) = 3)(.v and 5+ = (Sa) = 3)(a, where. v) and a) are the superposition states of the same form as Eqs. (107) and (108), the Hamiltonian (65) can be written as... 

Figure 13. Energy-level diagram of the superposition dressed states for A = fi. The solid lines indicate spontaneous transitions that occur independently of quantum interference, whereas the dashed lines indicate transitions that are significantly reduced by quantum interference.

So far we have concentrated attention on the vibrational manifold, without considering the rotational structure within this manifold. It should be noted at the outset that the rotational selection rules, AJ = 0, +1, valid for one-photon processes, completely break down for multiphoton processes, although the selection rules AM = 0, AK = 0 for a parallel transition remain valid. Thus, a wide range of J states associated with a particular vibrational state may become populated. Felker et al.22 have recently reported the observation of rotational coherence in large molecules. They observe a coherent superposition of precisely three J states, arising from AJ = 0, +1 in a one-photon process. The multiphoton process prepares a similar coherent population of J states, capable of exhibiting quantum interference phenomena, but many more J levels may be involved. 

Ci 2x, + IC2I X2- However, the individual probabilities do not add in this way for system in a quantum mechanical superposition state. Instead, the expectation value is C P - -C2 2 Ci +C2 P 2)- The classical and quantum mechanical predictions can be very different, because in addition to CipXi + C2pX2, the latter contains an interference term, C C2 Wi x P2) + C Ci( P2 x Pi). 

The path integral formulation builds on the principle of superposition, which leads to the celebrated quantum interference observed in the microscopic world. Thus, the amplitude for making a transition between two states is given by the sum of amplitudes along all possible paths connecting these states in the specified time, a concept familiar from wave... 

What has happened Note that before the transformation the states were indistinguishable by a state measurement of the classical basis 0) and (1) and afterwards they are as different as they could possibly be (i.e. orthogonal). We have now encountered the phenomenon of quantum interference. After the transformation (6.8) of state the amplitudes in front of basis state 1) have an opposite sign and cancel each other. These two terms in the superposition are said to have interfered destructively. On the other hand, the two terms with basis state 0) add up (i.e. they interfere constructively). The opposite happens with state J 2) after transformation (6.8). 

The initial state of the excited system has been represented as a superposition of the (time-independent) molecular eigenstates, each of which is a superposition of BO basis functions. The decay process is then described in terms of the time evolution of the amplitudes of the molecular eigenstates. The general theory of quantum mechanics implies that the decay of the state (10-4) will exhibit interference effects. 

The decay of the individual quasi-bound (metastable) resonance states follows an exponential law. The wave packet prepared by an ultrashort pulse can be represented as a (coherent) superposition of these states. The decay of the associated norm (i.e., population) follows a multi-exponential law with some superimposed oscillations due to quantum mechanical interference terms. The description given above is confirmed by experimental data. 

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