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Physical Review A

Nonlinear effects in quantum dissipation

David Vitali, Paolo Grigolini A

Published Dec 1, 1990• 21 citations
Publisher

Abstract

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)${\mathrm{〉}}_{\mathit{s}}$, where 〈${\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}$(t)${\mathrm{〉}}_{\mathit{s}}$ 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.

DOI
10.1103/physreva.42.7091

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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 \ensuremathω0{\mathrm{\ensuremath{\omega}}}_{0}\ensuremathω0​ 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 \ensuremathσx{\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}\ensuremathσx​(t)-〈\ensuremathσx{\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}\ensuremathσx​(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 [−−−BiologyandQuantumMechanics(Pergamon,Oxford,1982)]inthetwo−sitescase.ThismeansthatthenonlineareffectsprovenbyKenkreandco−workers[Phys.Rev.B34,4959(1986);35,1473(1987)]toaccompanyasignificantdepartureofthespinfromtheinitialstatearelostbytheNBA.Ontheotherhand,ourapproachprovidesarigorousevaluationoftheeffectsoftheoscillatorfluctuationsonthepredictionsoftheDNSE.ItisshownthatthequantumfluctuationsmighthaveasignificantrolealsointheregionofhightemperaturetowhichthenonadiabaticcorrectionsofKenkreandco−workersapply.Finally,itisshownthatastillmoreaccurateapproximationwouldbethatofneglectingareactionfieldproportionalto[---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 [−−−BiologyandQuantumMechanics(Pergamon,Oxford,1982)]inthetwo−sitescase.ThismeansthatthenonlineareffectsprovenbyKenkreandco−workers[Phys.Rev.B34,4959(1986);35,1473(1987)]toaccompanyasignificantdepartureofthespinfromtheinitialstatearelostbytheNBA.Ontheotherhand,ourapproachprovidesarigorousevaluationoftheeffectsoftheoscillatorfluctuationsonthepredictionsoftheDNSE.ItisshownthatthequantumfluctuationsmighthaveasignificantrolealsointheregionofhightemperaturetowhichthenonadiabaticcorrectionsofKenkreandco−workersapply.Finally,itisshownthatastillmoreaccurateapproximationwouldbethatofneglectingareactionfieldproportionalto{\mathrm{\ensuremath{\sigma}}}{\mathit{x}}(t)−〈(t)-〈(t)−〈{\mathrm{\ensuremath{\sigma}}}{\mathit{x}}(t)(t)(t){\mathrm{〉}}{\mathit{s}},where〈, where 〈,where〈{\mathrm{\ensuremath{\sigma}}}{\mathit{x}}(t)(t)(t){\mathrm{〉}}_{\mathit{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 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.
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.
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.
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.
Continuing our inquiry into the conditions when fluctuation-dissipation relations (FDR) may appear in the context of nonequilibrium dynamics of open quantum systems (over and beyond the conventional FDR from linear response theory) we turn to non-Gaussian systems and consider this issue for an anharmonic quantum oscillator interacting with a scalar quantum field bath. We present the general nonperturbative expressions for the rate of energy (power) exchange between the anharmonic oscillator and its thermal bath. For the cases that a stable final equilibrium state exists, and the nonstationary components of the two-point functions of the anharmonic oscillator have negligible contributions to the power balance, we can show nonperturbatively that equilibration implies an FDR for the anharmonic oscillator. We then use a weakly anharmonic oscillator as an example to illustrate the validity of those two assumptions and show that in the weak anhamonicity limit, they are satisfied according to our first-order perturbative results..
We use the numerical flow-equation renormalization method to study a nonlinear bath at low temperatures. The model of our nonlinear bath consists of a single two-level system coupled to a linear oscillator bath. The effects of this nonlinear bath are analyzed by coupling it to a spin, whose relaxational dynamics under the action of the bath is studied by calculating spin-spin correlation functions. As a first result, we derive flow equations for a general four-level system coupled to an oscillator bath, valid at low temperatures. We then treat the two-level system coupled to our nonlinear bath as a special case of the dissipative four-level system. We compare the effects of the nonlinear bath with those obtained using an effective linear bath, and study the differences between the two cases at low temperatures.