Superpositional state

As in classical mechanics, the outcome of time-dependent quantum dynamics and, in particular, the occurrence of IVR in polyatomic molecules, depends both on the Flamiltonian and the initial conditions, i.e. the initial quantum mechanical state I /(tQ)). We focus here on the time-dependent aspects of IVR, and in this case such initial conditions always correspond to the preparation, at a time of superposition states of molecular (spectroscopic) eigenstates involving at least two distinct vibrational energy levels. Strictly, IVR occurs if these levels involve at least two distinct... 

In a first time iirterval of the scheme (A3.13.46), a superposition state is prepared. This step... 

If now the nuclear coordinates are regarded as dynamical variables, rather than parameters, then in the vicinity of the intersection point, the energy eigenfunction, which is a combined electronic-nuclear wave function, will contain a superposition of the two adiabatic, superposition states, with nuclear... 

This spatial distribution is not stationary but evolves in time. So in this ease, one has a wavefunetion that is not a pure eigenstate of the Hamiltonian (one says that E is a superposition state or a non-stationary state) whose average energy remains eonstant (E=E2,i ap + El 2 bp) but whose spatial distribution ehanges with time. 

More generally, let the states of the ensemble be superposition states, or linear combinations of the eigenstates n > ... 

If the target is a pure electronic state (or coherent superposition state) in the conduction band, c)) the target operator can be written as... 

From a theoretical perspective, the object that is initially created in the excited state is a coherent superposition of all the wavefunctions encompassed by the broad frequency spread of the laser. Because the laser pulse is so short in comparison with the characteristic nuclear dynamical time scales of the motion, each excited wavefunction is prepared with a definite phase relation with respect to all the others in the superposition. It is this initial coherence and its rate of dissipation which determine all spectroscopic and collisional properties of the molecule as it evolves over a femtosecond time scale. For IBr, the nascent superposition state, or wavepacket, spreads and executes either periodic vibrational motion as it oscillates between the inner and outer turning points of the bound potential, or dissociates to form separated atoms, as indicated by the trajectories shown in Figure 1.3. 

The basic element of a quantum computer is the quantum bit or qubit. It is the QC counterpart of the Boolean bit, a classical physical system with two well-defined states. A material realization of a qubit is a quantum two-level system, with energy eigenstates, 0) and 1), and an energy gap AE, which can be in any arbitrary superposition cp) = cos(d/2) 0) + exp(i0)sin(0/2) l).These pure superposition states can be visualized by using a Bloch sphere representation (see Figure 7.1). 

Figure 9.11 Suppression of the 2P-1S spontaneous emission in the hydrogen atom, for which the natural linewidth is 1.66 X 10 cm". The solid lines display the decay of the optimized superposition of the Autler-Townes split levels with no interruptions. The dot-dashed lines are the decay curves of the same superposition states in the presence of interruptions. The dashed lines display the average decay of the two Autler-Townes split components. The optimization time t (marked by a triangle) is 0.2/T(= 0.65 ns), and the total time range displayed is up to 3/r(= 10 ns). A is the Autler-Townes splitting induced by the CW laser and T denotes the natural linewidth. Reprinted figure by permission from Ref. [38]. Copyright 2003 by the American Physical Society.

Figure 9.15 Decay dynamics for various initial vibrational states of the S2 electronic state, (a) Decay dynamics for initial 1> (0.53eV), 16> (0.76eV), 32> (0.86eV), 46> (0.92eV), and 64> (0.97 eV) states in the S2 space, (b) Decay dynamics of the initial superposition state used by Christopher et al. [30] with our methodology, when the 5 space includes vibrational states up to 1.4 and 1.6 eV. Reproduced with permission from Ref. [41]. Copyright 2010, AIP Publishing LLC.

A practical way of forming superposition states of correlated scattering states, thereby achieving control, is the topic of this chapter. Space limitations prevent more than a sketch details can be found elsewhere [5]. 

Control over the a, and production of the desired superposition states can be achieved by several routes. One nice way is to utilize the reactants from an earlier photodissociation step, altering the af by any of a number of coherent control scenarios [2] for this piereactive step. Consider then preparing n, 0) via a prereactive stage in which an adduct AB, made up of a structureless atom A and the molecular fragment B, is photodissociated. The AB is assumed to be initially in a pure state of energy Eg and the photodissociation is carried out with a coherent source. Under these circumstances photodissociation produces B in a linear combination of internal states. For... 

This then provides the desired 9 and x to ensure that the elements of the superposition state [Eq. (3)] are degenerate. 

The above formalism can be readily extended to general superposition states of the form... 

Suppose that the waves reaching some point in space originate from two i different sources. The principle of superposition states that the total wave displacement at that point is the sum of those caused by each individual wave. As a first example, consider the situation where a wave is reflected back on itself. The reflected wave has an equation similar to eqn 1.3, but with j the sign of x changed, corresponding to a reverse direction of travel. The combined (forward plus reflected) wave displacement is therefore... 

These devices allow utilizing new computational algorithms based on quantum superposition of states, allowing simultaneous representing many different numbers (so-called quantum computation). In a quantum computer information is loaded as a string of qubits (quantum mechanical representation of bits), which are quantum objects that can occupy different quantum states. A material implementation of qubits requires finding a medium, which can keep superpositional states from the destruction by interaction with the... 

Since the wavefunction is not a stationary state, it evolves according to Equation 6.11. If there are only two stationary states in the superposition state, as in Figure 6.3, the probability distribution and all of the observables oscillate at a frequency a> = (If — E )/Ti (see Problem 6-10). If we have the left side wavefunction in Figure 6.3 at time / = 0, at later times we will have a right side wavefunction. At... 

---