Quantum harmonic oscillator coherent states

The ground-state effective Hamiltonian is diagonal with eigenvalues ha n + 5], whereas the excited state one is that of a driven quantum harmonic oscillator that must lead to coherent states. 

Some Properties of Coherent States Expansion of the Coherent State on the Eigenvectors of the Quantum Harmonic Oscillator Hamiltonian... 

It is shown in Appendix N that the action of the translation operator on the ground state (0) ) of the Hamiltonian of the quantum harmonic oscillator gives a coherent state a ) ... 

N.l EXPANSION OF THE COHERENT STATE ON THE EIGENVECTORS OF THE QUANTUM HARMONIC OSCILLATOR HAMILTONIAN... 

By definition, a coherent state a) is the eigenvector of the non-Hermitean lowering operator a of the quantum harmonic oscillator. Thus, the basic equation and its conjugate are, respectively,... 

Now, we may insert in front of a coherent state, the closeness relation on the eigenstates of the number operator at a of the quantum harmonic oscillator (with [a, at] = 1), in the following way ... 

As a consequence, the expansion (N.2) of the coherent state on the eigenkets of the quantum harmonic oscillator Hamiltonian, gives... 

Next, first consider the average value of the coordinate of a quantum harmonic oscillator performed on a coherent state,... 

A quantum-mechanical treatment has been given for the coherent excitation and detection of excited-state molecular vibrations by optical absorption of ultrashort excitation and probe pulses [66]. Here we present a simplified classical-mechanical treatment that is sufficient to explain the central experimental observations. The excited-state vibrations are described as damped harmonic oscillations [i.e., by Eq. (11) with no driving term but with initial condition Q(0) < 0.] We consider the effects of coherent vibrational oscillations in Si on the optical density OD i at a single wavelength k within the Sq -> Si absorption spectrum. Due to absorption from Sq to Si and stimulated emission from Si and Sq,... 

This work is intended as an attempt to present two essentially different constructions of harmonic oscillator states in a FD Hilbert space. We propose some new definitions of the states and find their explicit forms in the Fock representation. For the convenience of the reader, we also bring together several known FD quantum-optical states, thus making our exposition more self-contained. We shall discuss FD coherent states, FD phase coherent states, FD displaced number states, FD Schrodinger cats, and FD squeezed vacuum. We shall show some intriguing properties of the states with the help of the discrete Wigner function. 

Figure 5.13 Quantum evolution of various initial coherent states of a truncated harmonic oscfllate, simulated by NMR (a) 0> (b) 0) + 2) (c) and (d) different oscillations of 0> + 1> + 2) + 3). Adapted with permission from [21].

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