We consider the spin-boson model as a measurement device, with the 1/2-spin system acting as a pointer and the influence of the environment simulated by means of the interaction between the 1/2-spin system and a dissipative oscillator. In accordance with the recent finding of Bonilla and Guinea [L. L. Bonilla and F. Guinea, Phys. Rev. A 45, 7718 (1992)] the assumption that the oscillator is classical results, under specific conditions, in the occurrence of a wave-function collapse. However, our theoretical analysis shows that the wave-function collapse is provoked by the action exerted by the 1/2-spin system on the oscillator (reaction field) and that the reaction field is as intense as the quantum-mechanical fluctuation neglected by the semiclassical approximation. A numerical treatment shows that no collapse occurs within a fully-quantum-mechanical treatment.
We study a two-level system linearly interacting with a set of quantum-mechanical oscillators, referred to as the ``bath.'' This system is formally equivalent to a magnetic dipole, with spin 1/2, precessing with the Larmor frequency ${\mathrm{\ensuremath{\omega}}}_{0}$ around a fixed magnetic field along the z axis and undergoing the influence of a fluctuating field along the x axis. These bath fluctuations are not independent of the state of the spin, and the crucial problem to be studied in this paper is precisely how to take the reaction field, i.e., the influence of the bath on the system, into account. We find a general result based on neglecting a contribution to the reaction field proportional to ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}$(t)-〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}$(t)〉, where the angle brackets denote averaging on both the spin and the bath space, with a density matrix corresponding to the spin polarized along the x axis. We show that under the special condition that the spin does not significantly depart from its initial state, our general result turns out to coincide with the noninteracting-blip approximation (NBA) of Leggett and co-workers [Rev. Mod. Phys. 59, 1 (1987)]. When we make the assumption that both quantum and thermal fluctuations of the bath can be neglected and that the time scale of the bath is virtually zero (adiabatic assumption), our general result turns out to coincide with the prediction of the discrete nonlinear Schr\"odinger equation (DNSE) of Davydov and Kislukha [Phys. Status Solidi B 59, 465 (1973)] and Davydov $[---Biology and Quantum Mechanics (Pergamon, Oxford, 1982)] in the two-sites case.This means that the nonlinear effects proven by Kenkre and co-workers [Phys. Rev. B 34, 4959 (1986); 35, 1473 (1987)] to accompany a significant departure of the spin from the initial state are lost by the NBA. On the other hand, our approach provides a rigorous evaluation of the effects of the oscillator fluctuations on the predictions of the DNSE. It is shown that the quantum fluctuations might have a significant role also in the region of high temperature to which the nonadiabatic corrections of Kenkre and co-workers apply. Finally, it is shown that a still more accurate approximation would be that of neglecting a reaction field proportional to ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}$(t)-〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}$(t)
〉s, where 〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}$(t)
〉s denotes an average carried out only on the spin space. This approximation leads our approach to coincide with the exact result in the special case where both the spin and its bath are replaced by their classical counterparts. Under this approximation, the detrapping would be equivalent to an Arrhenius-like thermal-activation process. When the coupling strength between the system and its bath is decreased, the spin is expected to depart from the initial trapped state. The dynamics of this process is expected to be influenced by the joint action of the bath fluctuations (both thermal and quantum mechanical) and of the nonlinearity stemming from the reaction field, and ignored by the NBA. This intriguing problem should be the subject of further investigations.
The authors study the semiclassical limit of the spin boson model, where the spin remains quantized and the field is expanded around the classical limit of a large number of photons, using a phase representation due to Bialynicki-Birula and Bialynicki-Birula (1976). They derive the semiclassical Hamiltonian and prove that in the semiclassical expansion there always arises a back reaction of the atom on the field. The nearest-neighbour spacing distribution of levels is studied in the same limit and shown to be, in general, qualitatively different from the distribution for the original model. However, the comparison between the Dyson-Mehta statistics is also studied and no qualitative difference is visible for the function Delta 3 .
We study a new model for treating quantum dissipative systems, in which the bath is modeled as a collection of spins coupled to the system of interest. We develop a quasiclassical method to study this model, approximating the quantum Heisenberg equations by the classical ones, supplemented with stochastic initial conditions carefully chosen so that the results obtained from the classical equations are as close as possible to the quantum results. Using this method we compare the dynamics of such a spin–spin system with that of a spin–boson system, in which the bath is modeled as a collection of harmonic oscillators. We verify numerically that when the system-bath coupling is spread over many bath spins (the Brownian motion limit), the spin–spin model can be mapped on the spin–boson model (although with a temperature dependent spectral density). We also demonstrate that the two dissipative models are qualitatively very different in a non-Brownian motion regime.
A semiclassical time-dependent self-consistent-field approach for the description of dissipative quantum phenomena is proposed. The total density operator is approximated by a semiclassical ansatz, which couples the system degrees of freedom to the bath degrees of freedom in a self-consistent manner, and is thus in the spirit of a classical-path description. The capability of the approach is demonstrated by comparing semiclassical calculations for a spin–boson model with an Ohmic bath to exact path-integral calculations. It is shown that the semiclassical model nicely reproduces the complex dissipative behavior of the spin–boson model for a large range of model parameters. The validity and accuracy of the semiclassical approach is discussed in some detail. It is shown that the method is essentially based on the assumption of complete randomization of nuclear phases. In particular, the assumption of phase randomization allows one to perform the trace over the bath variables through quasiclassical sampling of the nuclear initial conditions without invoking any further approximation.
We have studied a spin-1/2 system interacting with a classical stochastic oscillator in the overdamped regime, simulating a thermal bath. Our aim is to understand the influence of the reaction field, i.e., the effect of the dipole on its bath, which is usually neglected in the current treatments. We have faced the problem of the role of this reaction field with the help of computer simulation. We found that the trajectories of the dipole x component exhibit behaviors suggesting that, in some range of the parameters of the system, the model should have a bistable character. Moreover, the average equilibrium value of the dipole along the z axis shows a significant deviation from the ``weak-coupling predictions'' of either classical or quantum statistical mechanics. We have explained all these properties via a renormalization procedure which, in the strong-coupling regime, makes the Hamiltonian of interest very different from the bare Hamiltonian. Among other things, this renormalized Hamiltonian leads to a bimodal distribution for the energy, in excellent agreement with the numerical simulations.