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Physical Review A

Beyond the semiclassical approximation of the discrete nonlinear Schrödinger equation: Collapses and revivals as a sign of quantum fluctuations

Luca Bonci, Paolo Grigolini A, David Vitali

Published Oct 1, 1990• 23 citations
Publisher

Abstract

We study a two-state quantum-mechanical system, equivalent to a 1/2-spin dipole, with frequency ${\mathrm{\ensuremath{\omega}}}_{0}$, linearly interacting with a set of quantum oscillators. The corresponding discrete nonlinear Schr\"odinger equation (DNSE) has been given an analytical solution by Kenkre and Campbell [Phys. Rev. B 34, 4959 (1986)]. We show that in the strong-coupling limit their analytical expression is recovered from our microscopic approach when the thermal and quantum fluctuations of the bath of oscillators are neglected. To take into account the influence of thermal and quantum fluctuations on the prediction of the DNSE in the strong-coupling limit, we adopt a straightforward procedure based on the direct evaluation of 〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}$(t)〉, under the assumption alone that the frequency ${\mathrm{\ensuremath{\omega}}}_{0}$ is very weak. A special initial condition is used, with 〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}$(t)〉=1 and the oscillator at equilibrium in the corresponding effective potential. It is shown that in the strong-coupling limit this expression coincides with the noninteracting-blip approximation of Leggett et al. [Rev. Mod. Phys. 59, 1 (1987)] and the result of the equivalent projection approach of Aslangul, Pottier, and Saint-James [J. Phys. (Paris) 46, 2031 (1985); 47, 1657 (1986)]. Then, this expression is used to study the special case where the 1/2 spin interacts with only one oscillator at zero temperature. It is shown that the fast oscillations predicted by Kenkre and Campbell in the strong-coupling limit are damped by a Gaussian-like relaxation process provoked by the quantum fluctuations of the oscillator, which eventually lead to the destruction of the trapped state. These collapses are followed by periodical revivals reminiscent of those observed in quantum optics.

DOI
10.1103/physreva.42.4452

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We study a two-state quantum-mechanical system, equivalent to a 1/2-spin dipole, with frequency \ensuremathω0{\mathrm{\ensuremath{\omega}}}_{0}\ensuremathω0​, linearly interacting with a set of quantum oscillators. The corresponding discrete nonlinear Schr"odinger equation (DNSE) has been given an analytical solution by Kenkre and Campbell [Phys. Rev. B 34, 4959 (1986)]. We show that in the strong-coupling limit their analytical expression is recovered from our microscopic approach when the thermal and quantum fluctuations of the bath of oscillators are neglected. To take into account the influence of thermal and quantum fluctuations on the prediction of the DNSE in the strong-coupling limit, we adopt a straightforward procedure based on the direct evaluation of 〈\ensuremathσx{\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}\ensuremathσx​(t)〉, under the assumption alone that the frequency \ensuremathω0{\mathrm{\ensuremath{\omega}}}_{0}\ensuremathω0​ is very weak. A special initial condition is used, with 〈\ensuremathσx{\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}\ensuremathσx​(t)〉=1 and the oscillator at equilibrium in the corresponding effective potential. It is shown that in the strong-coupling limit this expression coincides with the noninteracting-blip approximation of Leggett et al. [Rev. Mod. Phys. 59, 1 (1987)] and the result of the equivalent projection approach of Aslangul, Pottier, and Saint-James [J. Phys. (Paris) 46, 2031 (1985); 47, 1657 (1986)]. Then, this expression is used to study the special case where the 1/2 spin interacts with only one oscillator at zero temperature. It is shown that the fast oscillations predicted by Kenkre and Campbell in the strong-coupling limit are damped by a Gaussian-like relaxation process provoked by the quantum fluctuations of the oscillator, which eventually lead to the destruction of the trapped state. These collapses are followed by periodical revivals reminiscent of those observed in quantum optics.
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{\mathrm{〉}}_{\mathit{s}}〉s​, where 〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}$(t)〉s{\mathrm{〉}}_{\mathit{s}}〉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.
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 spin-1/2 entity interacting with a set of quantum-mechanical oscillators using the Heisenberg picture. In the weak-coupling limit, the Heisenberg equation is given an analytical solution, suggesting that upon increase of the coupling the system should exhibit a critical transition to an overdamped regime, like that of a heavily damped oscillator. Thus the lifetime of the trapped state increases with increasing temperature. Under the assumption of very small tunneling frequency, another analytical solution is found, which predicts the occurrence of trapped states, whose lifetime decreases with an increase of temperature. As a result of the competition between the two trapping processes, the lifetime of the localized state is shown to be a nonmonotonic function of temperature with a minimum lifetime at a certain critical value of temperature. The predictions of this theoretical treatment are carefully checked by numerical solution of the interaction between a 1/2-spin entity and a dissipative oscillator, and a very satisfactory agreement between theory and numerical solution is found over the whole range of coupling strength.
We evaluate the time evolution of ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}$, the transverse component of a spin-1/2 dipole coupled to an Ohmic bath, using a theory balancing the quantum-mechanical fluctuations of the bath, as depicted by the noninteracting-blip approximation, with the nonlinear dynamics of the discrete nonlinear Schr\"odinger equation. As a relevant effect of the joint action of these two ingredients, we find that the time evolution of the system does not lead to the localization predicted by some equilibrium analyses. More precisely, starting from an eigenstate of ${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}$, the spin system will never asymptotically reach a broken-symmetry state.
Recent experiments on quantum behavior in microfabricated solid-state systems suggest tantalizing connections to quantum optics. Several of these experiments address the prototypical problem of cavity quantum electrodynamics: a two-level system coupled to a quantum harmonic oscillator. Such devices may allow the exploration of parameter regimes outside the near-resonance and weak-coupling assumptions of the ubiquitous rotating-wave approximation (RWA), necessitating other theoretical approaches. One such approach is an adiabatic approximation in the limit that the oscillator frequency is much larger than the characteristic frequency of the two-level system. A derivation of the approximation is presented, together with a discussion of its applicability in a system consisting of a Cooper-pair box coupled to a nanomechanical resonator. Within this approximation the time evolution of the two-level-system occupation probability is calculated using both thermal- and coherent-state initial conditions for the oscillator, focusing particularly on collapse and revival phenomena. For thermal-state initial conditions parameter regimes are found in which collapse and revival regions may be clearly distinguished, unlike the erratic evolution of the thermal-state RWA model. Coherent-state initial conditions lead to complex behavior, which exhibits sensitive dependence on the coupling strength and the initial amplitude of the oscillator state. One feature of the regime considered here is that closed-form evaluation of the time evolution may be carried out in the weak-coupling limit, which provides insight into the differences between the thermal- and coherent-state models. Finally, potential experimental observations in solid-state systems, particularly the Cooper-pair box---nanomechanical resonator system, are discussed and found to be promising.
The dynamics of a single quantum state embedded in one or several (quasi-)continua is one of the most studied phenomena in quantum mechanics. In this work we investigate its discrete analogue and consider short and long time dynamics based on numerical and analytical solutions of the Schrödinger equation. In addition to derivation of explicit conditions for initial exponential decay, it is shown that a recent model of this class [Phys. Rev. A 95, 053821, (2017)], describing a qubit coupled to a phonon reservoir with energy dependent coupling parameters is identical to a qubit interacting with a finite number of parallel regularly spaced band of states via constant couplings. As a consequence, the characteristic near periodic initial state revivals can be viewed as a transition of probability between different continua via the reviving initial state. Furthermore, the observation of polynomial decay of the reviving peaks is present in any system with constant and sufficiently strong coupling.