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Physical Review A

Localization breakdown as a joint effect of nonlinear and quantum dissipation

David Vitali, Luca Bonci, R. Mannella, Paolo Grigolini A

Published Feb 1, 1992• 16 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.45.2285

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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.

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Nonlinear effects in quantum dissipation

David Vitali, Paolo Grigolini A

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.

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We evaluate the time evolution of \ensuremathσx{\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}\ensuremathσx​, 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 \ensuremathσx{\mathrm{\ensuremath{\sigma}}}_{\mathit{x}}\ensuremathσx​, 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 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.
The spin-boson model, describing a two-level system coupled to a bosonic bath, is extensively studied as a paradigmatic dissipative quantum system, exhibiting rich dynamical behavior and even a localization transition in the strong coupling regime. Here, we additionally consider dephasing as a source of Markovian dissipation on top of the non-Markovian dynamics due to an Ohmic bath, and investigate the dynamics of the spin. We show that the characteristic frequency of the spin dynamics, while strongly renormalized by the bosonic bath, changes in a simple fashion (or does not change at all) with dephasing. To obtain these results, we develop an exact nonperturbative method known as the stochastic Schr\"odinger equation, mimicking the Ohmic bath via a stochastic magnetic field combined with the Lindblad quantum master equation due to dephasing, which allows us to numerically compute the dynamics. Furthermore, we derive weak-coupling analytic results utilizing the well-known noninteracting blip approximation. Our findings are relevant to quantum simulation of the spin-boson model in the regime of strong coupling in trapped ions and circuit QED architectures among others.
We explore how the dissipative geometric phase evolves within the spin-boson model, focusing specifically on coupling to an Ohmic bath in the weakly coherent regime. To determine the non-unitary time evolution of the system's reduced density matrix, we employ the non-interacting blip approximation (NIBA). For the localized pure initial state used throughout, we derive a compact Bloch-sphere expression showing that the mixed-state geometric phase is a weighted azimuthal winding of the dissipative trajectory. We then map geometric-phase accumulation across various system-bath coupling strengths, temperatures, static biases, and bath cutoff frequencies. Benchmarking representative results against the numerically exact time-evolving matrix product operator (TEMPO) technique shows that NIBA reproduces the population dynamics almost indistinguishably. It also captures the geometric phase quantitatively and qualitatively, although TEMPO reveals a clearer trend in the stationary coherence. Our results show that quantum dissipation suppresses the geometric phase through two complementary mechanisms: thermal noise reduces the state's purity, while static bias localizes the dynamics and reduces the accessible geometric area.
We consider the dissipative dynamics of a qubit coupled to a nonlinear oscillator (NO) embedded in an Ohmic environment. By treating the nonlinearity up to first order and applying Van Vleck perturbation theory up to second order in the qubit-NO coupling, we derive an analytical expression for the eigenstates and eigenfunctions of the coupled qubit-NO system beyond the rotating wave approximation. In the regime of weak coupling to the thermal bath, analytical expressions for the time evolution of the qubit's populations are derived: they describe a multiplicity of damped oscillations superposed to a complex relaxation part toward thermal equilibrium. The long-time dynamics is characterized by a single relaxation rate, which is maximal when the qubit is tuned to one of the resonances with the nonlinear oscillator.
We investigate the effect of dissipation in a Bose Josephson junction (BJJ) coupled to bath of bosons at two sites as an application of the generalized spin-boson model. Apart from the dynamical transition due to repulsive interactions, the BJJ undergoes a quantum phase transition by increasing the coupling strength with the bath modes. We analyze this system by mapping to an equivalent spin model coupled to the bosonic modes. The excitation energies and fluctuation of number imbalance are obtained within Holstein-Primakoff approximation, which exhibits vanishing of energy gap and enhanced quantum fluctuations at the critical coupling. We study the dynamics of BJJ using time-dependent variational method and analyze stability of different types of steady states. As a special case, we study in details the phase space dynamics of BJJ coupled to a single-mode, which reveals diffusive and incoherent behaviour with increasing coupling to the bath mode. The dynamical steady states corresponding to the Pi-oscillation and self-trapped state become unstable in the region where their oscillation frequencies are in resonance with the bath modes. We investigate the time evolution of number imbalance and relative phase in presence of Ohmic bath with Gaussian noise to incorporate thermal fluctuations. Apart from the damping of Josephson oscillations and transition to symmetry broken state for strong coupling, we observe the decay of Pi-oscillation and self-trapped state to the ground state as a result of dissipation. Variation of phase fluctuation with temperature of the bath shows similar behaviour as observed in the experiments.