By Joachim Ankerhold

In the final twenty years outstanding development has been made in knowing and describing tunneling methods in complicated structures when it comes to classical trajectories. This publication introduces contemporary recommendations and achievements with specific emphasis on a dynamical formula and kinfolk to express structures in mesoscopic, molecular, and atomic physics. complex instanton ideas, e.g. for decay premiums and tunnel splittings, are mentioned within the first half. the second one half covers present advancements for wave-packet tunneling in real-time, and the 3rd half describes thermodynamics and dynamical ways for barrier transmission in statistical, quite dissipative systems.

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**Example text**

1 there are always two trivial periodic orbits, namely, one at the well bottom q0 (σ) = 0 and one at the barrier top qb (σ) = qb . For temperatures T > T0 when no oscillating orbit in −V (q) exists (since hβ < 2π/ω0 ), these provide the dominating contributions to Z denoted by Z0 ¯ and Zb , respectively. It turns out that while around q0 ﬂuctuations y(σ) are stable, around qb there exists one unstable mode which induces translations in position around the barrier top. e. Z = Z0 + i|Zb |. 12). hβ > 2π/ω0 ), a new type of For suﬃciently low temperatures T < T0 (¯ periodic orbits appears which run in the time interval h ¯ β through the inverted 28 3 Tunneling in the Energy Domain barrier potential −V (q) with an arbitrary initial phase.

Hence, strings of instanton-antiinstanton pairs are also proper minimal action 44 3 Tunneling in the Energy Domain ( ) a -a Fig. 11. Multi-instanton orbit qn (σ) consisting of n = 6 individual instantons connecting the minima ±a of a bistable potential for ¯ hβ → ∞. paths obeying the boundary conditions [see Fig. 11]. ,σn (σ) , where n is the number of traversals and σ1 , . . , σn are the consecutive positions of the centers of the individual instantons/antiinstantons with the conhβ/2. Due to the boundary conditions for straint −¯ hβ/2 < σ1 < · · · < σn < ¯ paths contributing to ρβ (a, a) the number n must be even.

Presently, SQUIDS are used as elements for the implementation of superconducting quantum bits [39]. Fig. 12. Electron micrograph of a dc-SQUID ring with two JJs (outer ring) surrounding a SQUID ring containing three JJs (inner, darker ring). The outer SQUID is connected with two leads (top and bottom) carrying a bias current and the total circuit is threaded by a magnetic ﬂux. The white bar at the bottom represents a length scale of 5µm. Courtesy of A. Lupascu, ENS Paris. 25). The total ﬂux consists of the selfinduced ﬂux LI (self-inductance L) and an external ﬂux Φx according to Φ = Φx + LI.