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Journal de physique

Quantum ohmic dissipation : particle in an asymmetric double-well potential

Claude Aslangul, No�lle Pottier, Daniel Saint-James

Published Jan 1, 1986• 28 citations
Publisher

Abstract

The real-time dynamics of a particle in a slightly asymmetric double-well potential in the presence of ohmic dissipation is analysed by using standard quantum-statistical methods. At zero temperature, the particle eventually goes towards the lower potential well; the relaxation has an overall exponential character with possibly small oscillations; at finite temperature, the particle thermalizes over the two wells; two successive exponential regimes are observed in this case. The results about the final average position of the particle agree with the detailed balance principle and are in fact independent of the ohmic assumption. For an ohmic dissipation law, the localization phenomenon which occurs in the perfectly symmetric double-well potential at zero temperature disappears when an asymmetry is present, although a precursor behaviour is displayed for very low values of the asymmetry.

DOI
10.1051/jphys:01986004705075700

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The real-time dynamics of a particle in a slightly asymmetric double-well potential in the presence of ohmic dissipation is analysed by using standard quantum-statistical methods. At zero temperature, the particle eventually goes towards the lower potential well; the relaxation has an overall exponential character with possibly small oscillations; at finite temperature, the particle thermalizes over the two wells; two successive exponential regimes are observed in this case. The results about the final average position of the particle agree with the detailed balance principle and are in fact independent of the ohmic assumption. For an ohmic dissipation law, the localization phenomenon which occurs in the perfectly symmetric double-well potential at zero temperature disappears when an asymmetry is present, although a precursor behaviour is displayed for very low values of the asymmetry.
We study quantum tunneling in a biased double-well potential in the presence of Ohmic dissipation with a friction coefficient $\ensuremath{\alpha}>1$. The tunneling rate out of the metastable well is calculated in real time as a function of the temperature TTT and bias energy $\ensuremath{\epsilon}$. The T=0T=0T=0 result agrees with a droplet-model calculation in imaginary time. At low temperatures we find an enhancement to the T=0T=0T=0 rate varying as ${(\frac{T}{\ensuremath{\epsilon}})}^{2}$.
The quantum thermodynamic behavior of small systems is investigated in presence of finite quantum dissipation. We consider the archetype cases of a damped harmonic oscillator and a free quantum Brownian particle. A main finding is that quantum dissipation helps to ensure the validity of the Third Law. For the quantum oscillator, finite damping replaces the zero-coupling result of an exponential suppression of the specific heat at low temperatures by a power-law behavior. Rather intriguing is the behavior of the free quantum Brownian particle. In this case, quantum dissipation is able to restore the Third Law: Instead of being constant down to zero temperature, the specific heat now vanishes proportional to temperature with an amplitude that is inversely proportional to the ohmic dissipation strength. A distinct subtlety of finite quantum dissipation is the result that the various thermodynamic functions of the sub-system do not only depend on the dissipation strength but depend as well on the prescription employed in their definition.
The quantum thermodynamic behavior of small systems is investigated in presence of finite quantum dissipation. We consider the archetype cases of a damped harmonic oscillator and a free quantum Brownian particle. A main finding is that quantum dissipation helps to ensure the validity of the Third Law. For the quantum oscillator, finite damping replaces the zero-coupling result of an exponential suppression of the specific heat at low temperatures by a power-law behavior. Rather intriguing is the behavior of the free quantum Brownian particle. In this case, quantum dissipation is able to restore the Third Law: Instead of being constant down to zero temperature, the specific heat now vanishes proportional to temperature with an amplitude that is inversely proportional to the ohmic dissipation strength. A distinct subtlety of finite quantum dissipation is the result that the various thermodynamic functions of the sub-system do not only depend on the dissipation strength but depend as well on the prescription employed in their definition.
The problem of asymptotic-time symmetry breaking for a particle in a symmetric double-well, interacting with phonons, is formulated for a standard minimal two-site model and for initially asymmetric particle distribution and thermal but unrelaxed phonons. The timeconvolutionless Generalized Master Equations can be handled explicitly in the Born approximation. Formally extending the solution to the infinite time, the criteria for the asymptotic-time symmetry breaking comply with those known for the relaxed initial condition.
This chapter describes the analysis of the effect of dissipation on quantum properties of double-well systems. The simple qualitative observation suggests that interaction with a dissipative bath can significantly alter physical properties of a quantum mechanical particle in a double-well potential. The chapter provides the description of real-time dynamics of the double-well systems in the presence of linear Ohmic dissipation. It shows that dissipation yields total loss of coherence and, hence, real-time dynamics of the system remains incoherent at all times. Drastic simplifications occur if one assumes that the average time the system remains in a state described by the diagonal density matrix strongly exceeds the time it spends in the off-diagonal state. The collective charge variable is canonically conjugate to that of the phase, thus also describing quantum states of a macroscopically large number of electron degrees of freedom.
Dynamics of a two-level system with Ohmic dissipation driven by a time-dependent cw field in the noninteracting-blip approximation are studied. It is found that at low temperatures when \ensuremath{\alpha}, a friction parameter, lies between 1/2 and 1, the localization predicted by Chakravarty [Phys. Rev. Lett. 49, 681 (1982)] and Bray and Moore [Phys. Rev. Lett. 49, 1546 (1982)] disappears, and the rate constant for tunneling of the particle increases with the intensity and frequency of the field. For 01/2 quantum coherence is observed, and the transition temperature from the coherent to the decay regime depends on the field intensity. When the intensity parameter approaches the zeros of the zeroth Bessel function coherence disappears. This transition from coherent to incoherent motion oscillates with the intensity of the field, giving rise to a cascade of transitions at different temperatures.