Motional quantum state

To achieve both laser cooling and entan ement, we need to provide a coupling between intonal and motional quantum states. This can be achieved with the application of inhomogeneous (classical) electromagnetic fields. For example, consider an atom confined in a 1-D harmonic potential. The atom s dipole moment is assumed to couple to an electric field (x,t) through the Hamiltonian... 

The key to making a quantum logic gate is to provide conditional dynamics that is, we desire to perform on one physical subsystem a unitary transformation which is conditioned upon the quantum state of another subsystem [46]. In the context of cavity QED, the required conditional dynamics at the quantum level has recently been demonstrated [50,51]. For trapped ions, conditional dynamics at the quantum level has been demonstrated in verifications of zero-point laser cooling where absorption on the red sideband depended on the motional quantum state of the ion [11,12]. Recently, we have demonstrated a CN logic gate in this experiment, we also had the ability to prepare arbitrary input states to the gate (the keyboard operation of step (2a) below). 

I i i(q,01 in configuration space, e.g. as defined by the possible values of the position coordinates q. This motion is given by the time evolution of the wave fiinction i(q,t), defined as die projection ( q r(t)) of the time-dependent quantum state i i(t)) on configuration space. Since the quantum state is a complete description of the system, the wave packet defining the probability density can be viewed as the quantum mechanical counterpart of the classical distribution F(q- i t), p - P t)). The time dependence is obtained by solution of the time-dependent Schrodinger equation... 

Thus far, exaetly soluble model problems that represent one or more aspeets of an atom or moleeule s quantum-state strueture have been introdueed and solved. For example, eleetronie motion in polyenes was modeled by a partiele-in-a-box. The harmonie oseillator and rigid rotor were introdueed to model vibrational and rotational motion of a diatomie moleeule. 

From the point of view of the study of dynamics, the laser has three enormously important characteristics. Firstly, because of its potentially great time resolution, it can act as both the effector and the detector for dynamical processes on timescales as short as 10 - s. Secondly, due to its spectral resolution and brightness, the laser can be used to prepare large amounts of a selected quantum state of a molecule so that the chemical reactivity or other dynamical properties of that state may be studied. Finally, because of its coherence as a light source the laser may be used to create in an ensemble of molecules a coherent superposition of states wherein the phase relationships of the molecular and electronic motions are specified. The dynamics of the dephasing of the molecular ensemble may subsequently be determined. 

In the above relation, quantum states of phonons are characterized by the surface-parallel wave vector kg, whereas the rest of quantum numbers are indicated by a the latter account for the polarization of a quasi-particle and its motion in the surface-normal direction, and also implicitly reflect the arrangement of atoms in the crystal unit cell. A convenient representation like this allows us to immediately take advantage of the translational symmetry of the system in the surface-parallel direction so as to define an arbitrary Cartesian projection (onto the a axis) for the... 

The function g is the partition function for the transition state, and Qr is the product of the partition functions for the reactant molecules. The partition function essentially counts the number of ways that thermal energy can be stored in the various modes (translation, rotation, vibration, etc.) of a system of molecules, and is directly related to the number of quantum states available at each energy. This is related to the freedom of motion in the various modes. From equations 6.5-7 and -16, we see that the entropy change is related to the ratio of the partition functions ... 

Reaction dynamics deals with the intra- and intermolecular motions that characterize the elementary act of a chemical reaction. It also deals with the quantum states of the reactants and product. Since the dynamic study is concerned with the microscopic level and dynamic behaviour of reacting molecules, therefore, the term molecular dynamics is employed. 

In the adiabatic limit, the coupling F is strong, so that one may consider the transition between the two quantum states a continuous motion of the system on a single Bom-Oppenheimer surface (called the adiabatic state) that is the lowest eigenvalue of the 2 x 2 matrix in Eq. (18). 

Finally, the rules of angular momentum construction can be made as if the system had spherical symmetry. The reason is that the invariance to rotation of the I-frame leads to angular momentum conservation. Once all base states have been constructed, the dynamics is reflected on the quantum state that is a linear superposition on that base. As the amplitudes change in time, motion of different kinds result. 

At the most fundamental level one follows the time development of the system in detail. The reactants are started in a specific initial (quantum) state and the equation of motion are propagated to give the final state. The equation of motion of the system is the time dependent Schroinger equation, or, if the atoms involved are heavy enough (not H or Li) Newtons equation. The starting point is the adiabatic potential energy surface on which the process takes place. For some reactions electronic excitations during the reaction are important and must be included in addition to the electronically adiabatic dynamics. 

These expressions for the moments can be evaluated as equilibrium averages, without actually solving for all the quantum states of the system, or without solving the classical equations of motion for the classical trajectories, In the quantum-mechanical case, these equilibrium averages, Eq. (10), can be rewritten as traces, which can then be evaluated in any convenient basis. Thus the difficult step of solving for all the quantum states can be avoided in evaluating moments. 

Figure 14. (a) Potential-energy surfaces, with a trajectory showing the coherent vibrational motion as the diatom separates from the I atom. Two snapshots of the wavepacket motion (quantum molecular dynamics calculations) are shown for the same reaction at / = 0 and t = 600 fs. (b) Femtosecond dynamics of barrier reactions, IHgl system. Experimental observations of the vibrational (femtosecond) and rotational (picosecond) motions for the barrier (saddle-point transition state) descent, [IHgl] - Hgl(vib, rot) + I, are shown. The vibrational coherence in the reaction trajectories (oscillations) is observed in both polarizations of FTS. The rotational orientation can be seen in the decay of FTS spectra (parallel) and buildup of FTS (perpendicular) as the Hgl rotates during bond breakage (bottom). 

B. A. Hess The reason that macroscopic motions display coherence is that they are in most cases at the classical limit of quantum dynamics. In this case, a suitable occupation of quantum states ensures that quantum mechanical expectation values equal the classical value of an observable. In particular, the classical state of an electromagnetic field (the coherent state) is one in which the expectation value of the operator of the electromagnetic field equals the classical field strengths. 

I 2.1 Rotational Energy Levels of Diatomic Molecules, K I 2.2 Vibrational Energy Levels of Diatomic Molecules, 10 I 2.3 Electronic Stales of Diatomic Molecules, 11 I 2.4 Coupling of Rotation and Electronic Motion in Diatomic Molecules Hund s Coupling Cases, 12 1-3 Quantum States of Polyatomic Molecules, 14... 

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