Truncation scheme for double-well systems with Ohmic dissipation

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Abstract
In this paper we describe how a quantum system consisting of a single extended coordinate q in a double-well potential V(q), which interacts with a dissipative environment, may be systematically reduced (‘‘truncated’’) to an equivalent two-state system interacting with a modified environment. Although we concentrate on Ohmic dissipation, the method which we present is also applicable to other types of environmental spectral densities. The truncation scheme is applicable if there exists a wide separation of energy scales in the problem; we discuss the specific conditions which the system and environment must satisfy before such a scheme can be implemented. Our method proceeds by calculating the effects of the high-frequency environmental modes on the bare tunneling matrix element for transitions between the two wells. We are left with an equivalent two-state system with a renormalized tunneling matrix element, interacting with the remaining low-frequency environmental modes. The renormalized tunneling matrix element is calculated using path-integral techniques in a semiclassical approximation. Explicit results are given for a quartic double-well potential in the underdamped and overdamped regimes.