Canonical formulation of quantum tunneling with dissipation

Abstract

The tunneling in a double-well potential with dissipation is formulated without referring to instantons. This formulation, in the domain of small dissipation, goes beyond the instanton approximation. Several applications of this formulation for the Ohmic case are presented, and we show that the virtual mixing of excited states induced by dissipative interactions tends to enhance the tunneling rate. This enhancement effect becomes dominant for a super-Ohmic case. PACS numbers: 03.65.-w, 05.30.-d, 73.40.GkThe quantum tunneling (or coherence) in a double-well potential such as in Fig. 1 with dissipation received much attention recently [1-4]. In the conventional description of the phenomenon, which is based on the ground-state tunneling mediated by Euclidean instantons, it is not clear how the excited states in the double well affect the tunneling. In thisLetter, we present a canonical formulation of the quantum tunneling with dissipation. Our formulation, in the domain of small dissipation, goes beyond the instanton approximation and thus it can be used to describe macroscopic tunneling phenomena in a relatively shallow double well.We start with the second quantized Hamiltonian of the Caldeira-Leggett model [1],

Fluctuation-dissipation theorem and quantum tunneling with dissipation

“…These relations are identical to those in a field theoretical approach to the quantum tunneling with dissipation [7]. Our formula (25) is valid for the general potential problem , and it is not restricted to quantum tunneling with dissipation.…”

“…[7], where the unitarity and causality are explicitly incorporated. An explicit form of the Lagrangian contains more information and thus one can explicitly evaluate corrections to the energy eigenvalue in the second order of η, for example [7].…”

“…[7] that one obtains the energy splitting of lowest two levels in the deep double-well potential V (q) from (43) as (see also eq. (32))…”

“…We now specialize in the tunneling problem ( to be precise, quantum coherence) for which we have (isolated) nearly degenerate states arising from quantum tunneling . From the formula for E (1) n in ( 30), we obtain [7] …”

“…The present analysis is partly motivated by a field theoretical re-formulation [7] of the Caldeira-Leggett model, where the unitarity and causality are manifestly exhibited. In fact, the actual analysis of dissipative tunneling below closely follows that in Ref.…”