Classical coherent states

The generalized Prony analysis can extract a great variety of information from the ENDyne dynamics, such as the vibrational energy vib arrd the frequency for each normal mode. The classical quantum connection is then made via coherent states, such that, say, each nomral vibrational mode is represented by an evolving state... 

Electron Nuclear Dynamics (48) departs from a variational form where the state vector is both explicitly and implicitly time-dependent. A coherent state formulation for electron and nuclear motion is given and the relevant parameters are determined as functions of time from the Euler equations that define the stationary point of the functional. Yngve and his group have currently implemented the method for a determinantal electronic wave function and products of wave packets for the nuclei in the limit of zero width, a "classical" limit. Results are coming forth protons on methane (49), diatoms in laser fields (50), protons on water (51), and charge transfer (52) between oxygen and protons. 

Classical dynamics is studied as a special case by analyzing the Ehrenfest theorem, coherent states (16) and systems with quasi classical dynamics like the rigid rotor for molecules (17) and the oscillator (18) for various particle systems and for EM field in a laser. 

Abstract. The Dirac equation is discussed in a semiclassical context, with an emphasis on the separation of particles and anti-particles. Classical spin-orbit dynamics are obtained as the leading contribution to a semiclassical approximation of the quantum dynamics. In a second part the propagation of coherent states in general spin-orbit coupling problems is studied in two different semiclassical scenarios. 

In a second part we study the propagation of coherent states in general spin-orbit coupling problems with semiclassical means. This is done in two semiclassical scenarios h 0 with either spin quantum number s fixed (as above), or such that hs = S is fixed. In both cases, first approximate Hamiltonians are introduced that propagate coherent states exactly. The full Hamiltonians are then treated as perturbations of the approximate ones. The full quantum dynamics is seen to follow appropriate classical spin-orbit trajectories, with a semiclassical error of size yfh. As opposed to the first case,... 

The first case has already been considered section 2.0 the second case leads to a strong classical spin-orbit coupling, which is reflected in a Hamiltonian nature of the classical combined dynamics. In both situations the procedure is to find a suitable approximate Hamiltonian Hq( ) that propagates coherent states exactly along appropriate classical spin-orbit trajectories (x(l,),p(t),n(l,)). (For problems with only translational degrees of freedom this has been suggested in (Heller, 1975) and proven in (Combescure and Robert, 1997).) Then one treats the full Hamiltonian as a perturbation of the approximate one and calculates the full time evolution in quantum mechanical perturbation theory (via the Dyson series), i.e., one iterates the Duhamel formula... 

It is possible to improve the error term in (54) to any desired order hN/2 by replacing (45) with its N-th iterate. That way the state that is propagated along classical trajectories is no longer given by the coherent state (52), but by a suitably squeezed version of it. These states possess the same localisation properties as (52), however, their explicit form looks considerably more awkward. 

Any algebraic operator, written in terms of the boson operators a, n of Chapter 2 can be converted into a classical operator, written in terms of the variables (or p, q). We describe here again the derivation of van Roosmalen 1982. One introduces a group coherent state... 

The classical Hamiltonian, Hch is then the expectation value of the algebraic Hamiltonian, H, in the coherent state (7.66),... 

The general theory of classical limits of algebraic models is formulated not in terms of the group coherent states of Eq. (7.17) but rather in terms of projective coherent states. The ground-state projective coherent state is... 

Second, the mapping approach to nonadiabatic quantum dynamics is reviewed in Sections VI-VII. Based on an exact quantum-mechanical formulation, this approach allows us in several aspects to go beyond the scope of standard mixed quantum-classical methods. In particular, we study the classical phase space of a nonadiabatic system (including the discussion of vibronic periodic orbits) and the semiclassical description of nonadiabatic quantum mechanics via initial-value representations of the semiclassical propagator. The semiclassical spin-coherent state method and its close relation to the mapping approach is discussed in Section IX. Section X summarizes our results and concludes with some general remarks. 

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. 

Note that this dynamics is classical-like, as the coherent state properties density operator. 

Note that, as above, this classical-like dynamics is not without relation to the quasiclassical coherent state properties of the density operator involved in this average. 

The initial photon state can be a number state (with a not well-defined phase) or a linear combination of number states, for instance a coherent state. We formulate the construction of coherent states in the Floquet theory and show that choosing one as the initial photon state allows us to recover the usual semiclassical time dependent Schrodinger equation, with a classical held of a well-defined phase (see Section II.C). 

The choice of the value of the coherent state width 7 is arbitrary, since this parameter does not affect the mathematical properties of the Gaussian basis set which determine the form of the semi-classical propagator. In fact, the value of this quantity is usually chosen in practical implementations so as to facilitate the numerical convergence. In the following, we shall set 7 = 1/2 since with this choice (25) simplifies considerably and becomes... 

The most classical of all the quantum states is the coherent state. To compare the classical and quantum pictures, we therefore consider first the evolution of input wavepacket 0in) = i) 0 2) composed of the multimode coherent states cry) = Ylq ak) j = 1, 2). The states cry) are the eigenstates of the input... 

We have thus seen that the behavior of weak quantum fields is remarkably different from that of classical fields, as in the quantum regime the nonlinear phase shift is bounded between 27t tv2(i) 2/(L Sq). This fact severely limits the usefulness of weak coherent states for QI applications. Only in the limit of weak cross-phase modulation 9 [Pg.86]

To summarize, we have studied the interaction of two weak quantum fields with an optically dense medium of coherently driven four-level atoms in tripod configuration. We have presented a detailed semiclassical as well as quantum analysis of the system. The main conclusion that has emerged from this study is that optically dense vapors of tripod atoms are capable of realizing a novel regime of symmetric, extremely efficient nonlinear interaction of two multimode single-photon pulses, whereby the combined state of the system acquires a large conditional phase shift that can easily exceed 1r. Thus our scheme may pave the way to photon-based quantum information applications, such as deterministic all-optical quantum computation, dense coding and teleportation [Nielsen 2000]. We have also analyzed the behavior of the multimode coherent state and shown that the restriction on the classical correspondence of the coherent states severely limits their usefulness for QI applications. 

A Schrodinger cat-like state is a superposition of two macroscopically distinguishable classical states, [Schrodinger 1935 (a)], which for the harmonic oscillator are represented by strongly excited and sufficiently well separated (thus orthogonal) coherent states. To evolve a coherent state into a superposition, we may apply a unitary operator... 

In a series of papers, Nieto and colleagues examined in detail the problem of determining coherent states in general, and for some diatomic potential functions such as the Rosen-Morse, Morse, and Pdschl-Teller potentials (Nieto and Simmons, 1978, 1979 Nieto, 1978). The coherent-state approach is a powerful approach to study classically, i.e., at least intuitively, the energetic behavior of diatomics, and, as the authors suggest, provides the classical description of nuclear motion in diatomics. 

---