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Instantons Revisited: Dynamical Tunnelling and Resonant Tunnelling Jérémy Le Deunff, Amaury Mouchet

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Instantons Revisited: Dynamical Tunnelling and Resonant Tunnelling Jérémy Le Deunff, Amaury Mouchet Instantons revisited: dynamical tunnelling and resonant tunnelling Jérémy Le Deunff, Amaury Mouchet To cite this version: Jérémy Le Deunff, Amaury Mouchet. Instantons revisited: dynamical tunnelling and resonant tun- nelling. Physical Review E : Statistical, Nonlinear, and Soft Matter Physics, American Physical Soci- ety, 2010, 81, pp.046205. hal-00434131v2 HAL Id: hal-00434131 https://hal.archives-ouvertes.fr/hal-00434131v2 Submitted on 8 Apr 2010 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Instantons revisited: dynamical tunnelling and resonant tunnelling J´er´emy Le Deunff and Amaury Mouchet∗ Laboratoire de Math´ematiques et de Physique Th´eorique, Universit´eFran¸cois Rabelais de Tours — cnrs (umr 6083), F´ed´eration Denis Poisson, Parc de Grandmont 37200 Tours, France. (Dated: April 8, 2010) Starting from trace formulae for the tunnelling splittings (or decay rates) analytically continued in the complex time domain, we obtain explicit semiclassical expansions in terms of complex tra- jectories that are selected with appropriate complex-time paths. We show how this instanton-like approach, which takes advantage of an incomplete Wick rotation, accurately reproduces tunnelling effects not only in the usual double-well potential but also in situations where a pure Wick rotation is insufficient, for instance dynamical tunnelling or resonant tunnelling. Even though only one- dimensional autonomous Hamiltonian systems are quantitatively studied, we discuss the relevance of our method for multidimensional and/or chaotic tunnelling. PACS numbers: 05.45.Mt, 03.65.Sq, 03.65.Xp, 05.60.Gg, Keywords: Tunnelling, dynamical tunnelling, resonant tunnelling, instantons, semiclassical methods, Wick rotation, complex time I. INTRODUCTION: EVENTS OCCUR IN the original (non Wick-rotated) Hamilton’s equations — COMPLEX TIME may manifest itself through a transition that is not ne- cessarily a classically forbidden jump in position [4]. For Instantons generally refer to solutions of classical equa- instance, the reflection above an energy barrier, as a for- tions in the Euclidean space-time, i.e. once a Wick ro- bidden jump in momentum, is indeed a tunnelling pro- tation t it has been performed on time t in the cess [5, 6]. In the following, we will consider the case Minkowskian7→ − space-time. Since the mid-seventies, they of a simple pendulum whose dynamics is governed by have been extensively used in gauge field theories to de- the potential V (q) = γ cos q; for energies larger than scribe tunnelling between degenerate vacua [1]. In intro- the strength γ > 0 of the potential, one can observe a ductory texts [2, 3, for instance], they are first presented quantum transition between states rotating in opposite within the framework of quantum mechanics: when the direction while two distinct rotational classical solutions, classical Hamiltonian of a system with one degree of free- obtained one from each other by the reflection symmetry, dom has the usual form are always disconnected in real phase-space. 1 H(p, q)= p2 + V (q) (1) A second example is provided by a typical situation of 2 resonant tunnelling: when, for instance, V has a third, (p and q denote the canonically conjugate variables), the deeper, well which lies in between two symmetric wells Wick rotation induces an inversion of the potential and, (see Fig. 6a), the oscillation frequency between the latter then, some classical real solutions driven by the trans- can be affected by several orders of magnitude, when two formed Hamiltonian p2/2 V (q) can be exploited to eigenenergies get nearly degenerate with a third one, cor- quantitatively describe a tunnelling− transition. As far responding to a state localised in the central well. Then, as we know, in this context, only the simplest situa- we lose the customary exponential weakness of tunnelling tions have been considered, namely the tunnelling de- and it is worth stressing, coming back for a second to cay from an isolated minimum of V to a continuum and quantum field theory, that having a nearly full tunnelling the tunnelling oscillations between N degenerated mi- transmission through a double barrier may have drastic nima of V that are related by an N-fold symmetry. In consequences in some cosmological models. In this exam- ple, one can immediately see (Fig. 6b) that V also has those cases, what can be captured is tunnelling at the − lowest energy only. However, not to speak of the highly an energy barrier and working with a complete Wick ro- non-trivial cases of tunnelling in non-autonomous and/or tation only remains insufficient in that case. non-separable multidimensional systems, there are many situations that cannot be straightforwardly treated with In order to describe the tunnelling transmission at an a simple inversion of the 1d, time-independent, potential. energy E below the top of an energy barrier, which may First, tunnelling — i.e. any quantum phenomenon be crucial in some chemical reactions, the pioneer works that cannot be described by real classical solutions of by Freed [7], George and Miller [8, 9], have shown that the computation of a Green’s function G˜(qf , qi; E) (or a scattering matrix element) requires taking into consider- ation classical trajectories with a complex time. These ∗email: jeremy.ledeunff@lmpt.univ-tours.fr, complex times come out when looking for the saddle- [email protected] point main contributions to the Fourier transform of the 2 time propagator, whose discrete energy spectrum and the associated or- thonormal eigenbasis are defined by 1 ∞ iEt/~ G˜(qf , qi; E)= G(qf , qi; t)e dt , (2) √2π~ Hˆ φ± = E± φ± ; Sˆ φ± = φ± (4) Z0 | n i n | n i | n i ± | n i which is, up to now, the common step shared by all where n is a natural integer. When the Planck cons- the approaches involving complex time [10–14, for ins- tant ~ is small compared to the typical classical actions, tance]. Though well-suited for the study of scattering, standard semi-classical analysis [22–24] shows that one indirect computations are required to extract from the can associate some classical regions in phase-space to poles of the energy Green’s function (2) some spectral each eigenstate φn± . This can be done by construc- signatures of tunnelling in bounded systems. Generically, | i ting a phase-space representation of φn± , typically the these signatures appear as small splittings between two Wigner or the Husimi representation, and| i look where the quasi-degenerate energy levels and can be seen as a nar- corresponding phase-space function φn±(p, q) is mainly row avoided crossing of the two levels when a classical localised (all the more sharply than ~ is small). For parameter is varied [15–18]. In the present article, we a Hamiltonian of the form (1) where V has local mi- propose a unified treatment that provides a direct com- nima, some of the eigenstates remain localised in the putation of these splittings (formulae 9 or 10); as shown neighbourhood of the stable equilibrium points. For ins- in section II, it takes full advantage of the possibility of tance, for a double-well potential whose shape is shown working not necessarily with purely imaginary time, but + in Fig. 1a), the symmetric state φn and the antisym- with a general parametrisation of complex time as first | i metric state φn− at energies below the local maximum suggested in [19]. The semiclassical approach naturally of the barrier| havei their Husimi representation localised follows (sec. III) and some general asymptotic expansions around both stable equilibrium points (p, q) = (0, a), can be written (equations (C6) and (40)) and simplified more precisely along the lines H(p, q) = E (1d tori)± at (equation (73)); they constitute the main results of this + energy E En En−. Because of the non-exact de- paper. To understand where these formulae come from ≃ + ≃ + generacy of En and En−, any linear combination of φn and how they work, we will start with the paradigmatic | i and φn− constructed in order to be localised in one well case of the double-well potential (sec. IV) and the sim- only| willi not be stationary anymore and will oscillate ple pendulum (sec.V); we defer some general and techni- back and forth between a and a at a tunnelling fre- cal justifications in the appendices. Then we will treat + ~ − quency En− En /(2π ). At low energy this process the resonant case in detail in sec. VI, where an appro- is classically| − forbidden| by the energy barrier. This is iθ priate incomplete Wick rotation t e− t provides the very general, even for multidimensional non-integrable key to showing how interference effects7→ `ala Fabry-P´erot + systems: the splitting ∆En = En− En between some between several complex trajectories reproduce the non- nearly degenerated doublets provides| − a quantitative| ma- exponential behaviour of resonant tunnelling, already at nifestation of the tunnelling between the phase-space re- work in open systems with a double barrier [20, 21]. After gions where the corresponding eigenstates are localised. having shown how to adapt our method to the computa- Rather than computing the poles of E G˜(a, a, E), tions of escape rates from a stable island in phase-space another systematic strategy to obtain7→ one individual− (sec. VII), we will conclude with more long–term con- splitting [48] is to start with Herring’s formula [25–27] siderations by explaining how our approach provides a that relies on the knowledge of the eigenfunctions outside natural and new starting point for studying tunnelling in the classically allowed regions in phase-space.
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