Dissipative tunnelling by means of scaled trajectories

P. V. Krevchik, Кревчик Владимир Дмитриевич, Михаил Борисович Семенов

Luis Ratia

The Ratia postulate proposes that wave function collapse is triggered when energydissipated from a quantum system into its environment satisfies a Heisenberg–Landauer-typeaction threshold, ∆Edis · τcollapse ≥ ℏ/2. This note derives a falsifiable structural criterion forwhen this threshold would be crossed during a dissipative quantum tunneling event, usingthe well-established framework of Grabert, Weiss, and Hanggi (Phys. Rev. Lett. 52, 2193,1984) for tunneling out of a metastable state coupled to an Ohmic thermal bath. Usingonly quantities already defined in that framework — the friction coefficient η, the barrierwidth q0, and the "bounce length" τB — we show that the postulate’s threshold condition,evaluated over the characteristic duration of the tunneling bounce trajectory, reduces to acompact dimensionless criterion, R = 4α (M ω0q20 /ℏ) ≷ 1, independent of τB itself. Thiscriterion is dimensionally verified and shown to coincide, up to a numerical prefactor, withthe dissipative tunneling literature’s own characteristic action scale. We discuss its physicalinterpretation, its qualitative behavior across parameter regimes, and its current limitations— principally the absence of a numerically evaluated real physical system, which we identifyas the necessary next step.

A. O. Caldeira

N. Emir Anuar

We present a formalism for which a dissipative system is given by a variational principle. The formalism applies to dynamical systems where its trajectory is monotonic. Subsequently, we derive its Lagrangian and Hamiltonian. From the Hamiltonian, we quantize canonically the classical particle in a viscous media. We study the free quantum particle in a viscous media and the dissipative quantum tunnelling. It is found that the dissipation influences tunnelling probability by a factor that closely resembles the result of Caldeira and Leggett.

S. V. Mousavi

Diffraction in time and early-arrival phenomena will be studied in the context of dissipative systems. This theoretical consideration is done by using the linear scaled wave equation proposed in the framework of Caldirola-Kanai theory for the continuous quantum-classical transition. Two simple but physically important and interesting examples are studied, sudden release from a shutter where transient behaviors (diffraction in time) are seen; and also transmission through a time-dependent parabolic barrier where early-arrival is seen. Calculations show that diffraction in time is a non-classical effect and temporary behaviors are gradually suppressed as the friction increases. Furthermore, early-arrival is seen even in the classical regime. This means that early-arrival is not a non-classical phenomenon. This behavior will be explained by means of scaled trajectories which are just Bohmian trajectories in the quantum regime.