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
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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. Here, we multidimensional systems. will propose alternative formulae [28] that involve traces of a product of operators, among them the evolution ope- rator, II. TUNNELLING SPLITTINGS
def iHT/ˆ ~ Uˆ(T ) = e− A particularly simple signature of tunnelling can be ∞ + − identified when the Hamiltonian has a two fold sym- + + iE T/~ iE T/~ = φ φ e− n + φ− φ− e− n , (5) metry and, therefore, we will consider quantum systems | n i h n | | n i h n | n=0 whose time-independent Hamiltonian Hˆ = H(ˆp, qˆ) com- X mute with an operator Sˆ such that Sˆ2 = 1 (the ˆ allows analytically continued in some sector of the complex to distinguish the quantum operators from the classical time domain. This approach, which privileges the time- phase-space functions or maps). In most cases, Sˆ stands domain, provides, of course, a natural starting point for a for the parity operator: semiclassical analysis in terms of classical complex orbits. The simplest situation occurs when the tunnelling dou- + H( p, q)= H(p, q) . (3) blet is made of the two lowest energies E0 . E0− = − − + ~ E0 +∆E0 of the spectrum. If ω denotes the energy dif- ˆ Sˆ The spectrum of H can be classified according to and, ference between the nearest excited states and E0± (for for simplicity, we will always consider a bounded system the double well potential E± E± + ~ω where ω is the 1 ≃ 0 3 classical frequency around the stable equilibrium points), specific prescriptions on the choice of T , in order to im- by giving a sufficiently large imaginary part to T , prove the accuracy of ∆En. We will illustrate how this − works in the examples of secs. IV, V and VI. ω Im(T ) 1 , (6) − ≫ Let us mention another way to select excited states, that we did not exploit further. With the help of a posi- we can safely retain the n = 0 terms only, which exponen- tive smooth function F (u) that has a deep, isolated min- tially dominate the trace of (5). To be valid, this approx- imum at u = 0, say F (u)= u2N with N strictly positive, imation requires that we remain away from a quantum we can freeze the dynamics around any energy E by con- resonance where the definition of the tunnelling doublet def sidering a new Hamiltonian H′(p, q) =F H(p, q) E) . is made ambiguous by the presence of a third energy level − + Classically, the phase-space portrait is the same as the in the neighborhood of E0 and E0−. Then we have im- original one obtained with Hamiltonian H except that mediately the set of points H(p, q)= E now consists of equilibrium def points. The quantum Hamiltonian Hˆ ′ =F (Hˆ E) has T ∆E tr Sˆ Uˆ(T ) − tan 0 i . (7) the same eigenfunctions as Hˆ but the corresponding spec- 2~ ≃− tr Uˆ(T ) + trum is now F (En± E). By choosing E = En , the doublet E yields to− the ground-state doublet ∆E = ~ n± n′ When the tunnelling splitting is smaller than ω by sev- F (∆E ) and we can use the approximation (9) with Uˆ = eral orders of magnitude, we can work with a complex n ′ exp( iHˆ ′T ). With the F given above, we have time such that − 1/(2N) T ∆E0 ~ Sˆ ˆ | | 1 (8) 1/(2N) def 2 tr U ′(T ) ~ ∆E [∆′ (T )] = . 2 ≪ n ≃ n iT ˆ " tr U ′(T ) # remains compatible with condition (6) and, therefore, (11) 2~ tr Sˆ Uˆ(T ) ∆E ∆ (T ) def= (9) 0 0 ˆ III. SEMICLASSICAL EXPRESSIONS ≃ iT tr U(T ) will provide a good approximation of the tunnelling split- A. Hamiltonian dynamics with complex time ting for a wide range of T . Though condition (8) is widely fulfilled in many situations this is not an essential condi- Formally, the numerator and the denominator of the tion since one can keep working with tan 1. Numerically, − right hand side of (9) can be written as a phase-space the estimation (9) has also the advantage of obviating a path integral of the form diagonalization. When we want to compute a splitting, due to tun- ~ nelling, between an arbitrary doublet, the selection of eiS[p,q;t]/ D[p]D[q] (12) the corresponding terms in the right hand side of (5) ZP ˆ can be made with an operator Πn that will mimic the where the continuous action is the functional + + (a priori unknown) projector φn φn + φn− φn− . It | i h | | i h | sf will be chosen such that its matrix elements are localised def dq dt S[p, q; t] = p(s) (s) H p(s), q(s) (s) ds. in the regions of phase-space where φn±(p, q) are domi- ds − ds ~ Zsi nant. Under the soft condition T ∆En/(2 ) 1, we (13) | | ≪ will therefore take The subset of phase-space paths, the measure D[p]D[q] and the actionP (13) appear as a continuous limit (τ 0) 2~ tr Sˆ Πˆn Uˆ(T ) → ∆E ∆ (T ) def= . (10) of a discretized expression whose precise definition de- n n ˆ ˆ ≃ iT tr ΠnU(T ) pends on the choice of the basis for computing the traces but, in any cases, involves a typical, finite, complex time The localization condition on the matrix elements of Πˆn step τ (see also appendix A). The complex continuous is a selection tool that replaces (6). In that case, there is time path s t(s) is given with fixed ends t(s )= t = 0, 7→ i i a battle of exponentials between the exponentially small t(sf )= tf = T . Because the slicing of T in small complex ˆ matrix elements φm± Πn φm± and the time dependent time steps of modulus of order τ is arbitrary, the inte- i(Em En)hT/~ | | i terms e− − . The last term would eventually grals of the form (12) remain independent of the choice dominate for the lower energy states (Em < En) if Im T of t(s) for si 2 dp ∂H dt κo i ∂ So iSo/~ = ; (14a) G(qf , qi; T ) ( 1) det e . ds − ∂q ds ~∼0 − 2π~ ∂qi∂qf → o s dq ∂H dt X (16) = , (14b) ds ∂p ds The sum involves (complex) classical trajectories o in phase-space i.e. solutions of equations (15) with q(si)= with appropriate boundary conditions imposed on some qi, q(sf ) = qf for a given [t] such that t(si) = 0, o canonical variables at si and/or sf . When H is an an- t(sf )= T . The action So is computed along with defi- alytic function of the phase-space coordinates (p, q), we nition (13) and is considered as a function of (qf , qi; T ). can take the real and imaginary parts of equations (14), The integer κo encapsulates the choice of the Riemann use the Cauchy-Riemann equations that render explicit sheet where the square root is computed; it keeps a record the entanglement between the real part and the imagi- of the number of points on o where the semiclassical ap- nary part of any analytic function f(z) : Re(df/dz) = proximation (16) fails. As far as we do not cross a bifurca- ∂(Re f)/∂(Re z) = ∂(Im f)/∂(Im z) and Im(df/dz) = tion of classical trajectories when smoothly deforming [t], ∂(Im f)/∂(Re z)= ∂(Re f)/∂(Im z), and then obtain the number of o’s, the value of So and κo do not depend − of the choice of [t]. The numerator (resp. denomina- d Re p ∂ dt tor) of ∆0(T ) are given by the integral G(ηq,q; T )dq = Re H ; (15a) where η = 1 (resp. η = +1). Within the semiclassi- ds −∂ Re q ds cal approximation,− when o contributes toR the propaga- d( Im p) ∂ dt − = Re H ; (15b) tor G(ηq,q; T ), we have to evaluate ds −∂ Im q ds 2 d Re q ∂ dt i ∂ So ~ = Re H ; (15c) ( 1)κo dq det eiSo(ηq,q,T )/ . ds ∂ Re p ds − v 2π~ ∂q ∂q Z u i f (ηq,q;T )! d Im q ∂ dt u = Re H . (15d) t (17) ds ∂( Im p) ds The steepest descent method requires the determination − of the critical points of So(ηq,q,T ). Since the momenta Therefore, the dynamics described in terms of complex at the end points of o are given by canonical variables is equivalent to a dynamics that re- p = p(s ) = ∂ So(q , q ; T ); (18a) mains Hamiltonian — though not autonomous with re- i i − qi f i spect to the parametrisation s whenever dt/ds varies pf = p(sf ) = ∂qf So(qf , qi; T ) , (18b) with s —, involving twice as many degrees of freedom as the original system, namely (Re q, Im q) and their re- the dominant contributions to tr Uˆ(T ) come from perio- spectively conjugated momenta (Re p, Im p). The new dic orbits, i.e. when (pf , qf ) = (pi, qi), whereas the dom- − Hamiltonian function is then Re(Hdt/ds) but it would inant contributions to tr SˆUˆ(T ) come from half symme- have been equivalent (though not canonically equivalent) tric periodic orbits, i.e. when (pf , qf ) = ( pi, qi). As to choose the other constant of motion Im(Hdt/ds) asa explained in detail in the appendix A, one− must− distin- Hamiltonian. guish the contributions of the zero length orbits e (the What will be important in what follows is that [t] equilibrium points) from the non-zero length periodic or- will not be given a priori. Unlike in the standard in- bits o. For one degree of freedom, their respective con- stanton approach where t is forced to remain on the tributions are given, up to sign, by imaginary axis, we will see that for describing tunnelling ~ e iH(pe,qe)T/ it is a more efficient strategy to look for some com- − , (19a) λeT/2 λeT/2 plex paths p(s), q(s) that naturally connect two phase- e ηe− − space regions and then deduce [t] from one of the equa- with λe the two Lyapunov exponents of the equilibrium tions (14). It happens that in the two usual textbook point±e = (p , q ), and examples that we mentioned in the first paragraph of e e our Introduction, the complex time [t] has a vanishing β Tβ dEo ~ real part, but in more general cases where tunnelling be- ( 1)µo eiSo/ , (19b) − √ 2ηiπ~ dT tween excited states is studied, this is no longer true. P− r where o is a periodic orbit of period To = T if η = +1 and half a symmetric periodic orbit of half period To = T if η = 1. The sum runs over all the branches β that B. Trace formulae − compose the geometrical set of points belonging to o. Tβ is the characteristic time (A12) on the branch β. The Let us privilege the q-representation and consider the energy Eo is implicitly defined by (A10) and µo essen- analytic continuation for complex time T of the well- tially counts the number of turning points on o. 5 Expressions (19) are purely geometric; their classical In general we will not be able to prove that other com- ingredients do not rely on a specific choice of canonical plex periodic orbits give sub-dominant contributions but coordinates and they are independent of the choice of the the examples given in the following sections are rather basis to evaluate the traces. convincing. Moreover, our criterion of selection allows In the general case, the most difficult part consists in us to justify the four rules presented in [31, sec. IIA] determining which periodic orbits contribute to the semi- for computing the contributions of complex orbits to the classical approximation of the traces. Condition To = T semiclassical expansions of the energy Green’s function. is necessary but far from being sufficient; the structure of the complex paths (keeping a real time) may appear to be very subtle [29] and have been the subject of many IV. APPLICATION TO THE DOUBLE-WELL recent delicate works [30]. Even for a simple oscillating POTENTIAL integral, the determination of the complex critical points of the phase that do contribute is a highly non-trivial Let us show first that our strategy leads to the usual in- problem because it requires a global analysis: one must stanton results for a Hamiltonian of the form (1) with an know how to deform the whole initial contour of integra- even V having two stable symmetric equilibrium points tion to reach a steepest descent paths. at q = a (Fig. 1a); in their neighbourhood, the fre- ± Our strategy consists in retaining the terms (19b) for quency of the small vibrations is ω. We are interested ~ which we can choose the complex time [t] to constrain in the ground-state splitting ∆E0 for small enough in ~ the (half) periodic orbit o to keep one of the canonical order to have ∆E0 ω. ≪ coordinate (say q) real. This o surely contributes because Before we make any semiclassical approximations, we we do not have to deform the q part of the integration do- check in Fig. 2 the validity of the estimation (9) on the main of (12) in the complex plane. But we will see in the quartic potential by a) verifying that ∆0(T ) is almost next section that two different periodic orbits with real q real and independent of the choice of the complex T pro- correspond to two different choices of [t]: For a chosen [t], vided that conditions (6) and (8) are fulfilled, and b) by only some isolated points, if any, in phase space, will pro- checking that this constant gives a good approximation vide a starting point (pi, qi) of a trajectory with q(s) real of the “exact” ∆E0 computed by direct numerical diag- all along o. When sliding slightly the initial real coor- onalization of the Hamiltonian [32]. dinate qi in the integral (17), it requires to change [t] As explained at the end of section sec. III, in phase- as well to maintain q(s) real on the whole o. This is space we will try to find some time-path [t] that allows not a problem since all the quantities involved in expres- the existence a (half) symmetric periodic orbit o that sion (17) are [t]-independent if the deformation of [t] is connects two tori at (real) energy E & 0 in the neigh- small enough not to provoke a bifurcation of o, that is, bourhood of (p, q) = (0, a) while q remains real. If we ± whenever the initial point does dot cross one of the turn- impose t(si) = 0 and t(sf ) = T , then E = Eo(qf , qi,T ) ing points, which are the boundaries of the branches β. is implicitly given by relation (A10). We will denote The integral (17) may be computed with a fixed time- by qr(E) (resp. qr′ (E)) the position of the turning path or an adaptive one for each separated branch. This point at energy E > 0 that lies in between q = 0 computation, presented in the appendix A, generates the and q = a (resp. that is larger that a). The two sum over all branches that appears in formula (19b). To branches p (q, E) = 2 E V (q) are either purely sum up, to keep a contribution to the traces, we must real when V±(q) > E or± purely imaginary− when V (q) 6 E. know if we can choose a shape of [t] in order to pick a Then, from equation (14b),p real-q periodic orbit with a specific q . i dt 1 dq This construction is certainly not unique (one may = (21) choose other constraints) so we will use the intuitive prin- ds p ds ciple that the periodic orbits we choose will connect the is purely real in the classically allowed region, while two regions of phase-space that are concerned by tun- purely imaginary in the forbidden region. Therefore the nelling i.e. where φn± are dominant; this is justified by complex time path [t] must have the shape of a descend- the presence of the operator Πˆn in the right hand side ing staircase whose steps are made of pure real or pure def of (10). The prefactor Πn(q, q′) = q Πˆn q′ will select imaginary variations of time (see Fig. 3). in the integral h | | i The complex orbit with real q can be represented in phase-space as a continuous concatenation of paths, fol- lowing the lines E = (Re p)2/2+ V q) in the allowed dq dq Π (q, q ) G(ηq , q; T ) , (20) ′ n ′ ′ region and E = (Im p)2/2 V (q) in the forbidden re- Z gion. It is natural− to represent− o in the three dimensional a domain around the projection onto the q-space of the section Im q = 0 of the complex phase-space with axes appropriate tori. In the specific case of the ground-state given by (Re q, Re p, Im p): the junctions at the turning splitting, condition (6) does the job of Πˆ0: the large points lie necessarily on the (Re p =0, Im p = 0) axis (see value of Im T requires that the orbit approach at least Figs. 1c and d). A periodic orbit is made of a succession one equilibrium point and then follow a separatrix line. of repetitions of 6 a) | 0 Re ∆ | Re T − Im T b) 0.25 0 -0.25 -0.5 -0.75 -1 -1.25 -1.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1/~ Figure 1: (colour online) In the case of a double well potential shown in a), tunnelling can be described by a solution of (14) Figure 2: (colour online) For V (q)=(q2 a2)2 with a = 1 with a purely real q evolving from one well to the other. In and ~ 1/12, we have plotted in the upper− graph a) the real ≃ phase-space, this trajectory appears to be a concatenation of part of ∆0 defined by (9) as a function of the complex T . two types of curves that join on the (Re q)-axis: 1) a trajec- It becomes constant for Im T large enough for having (6) tory that lies in the phase-space plane (Re q, Re p) with real (1/ω = 1/(2√2a) .35) regardless of Re T . In the same ≃ −10 variations of t at energy E with the potential V and 2) a range, Im ∆0(T )) is negligible compared to ∆E0 4.4 10 trajectory that lies in the phase-space plane (Re q, Im p) with and condition (8) is largely fulfilled. In the lower graph∼ b), we imaginary variations of t at energy E with the potential V have computed the “exact” value ∆E0 by direct diagonaliza- − − shown in b). In c) a family of constant energy curves is shown tion. The black thick solid line corresponds to ln(∆E0) from def (horizontal green (light gray) for the first type, vertical red which we have substracted Λ(~) = S˜c(~ω/2)/(2~)+ln(~ω/π) (dark gray) for the second type). In d) for a given energy, we in order to emphasize the contribution− of the prefactor. The l c r show how the three curves , , glue together at the turning blue solid line corresponds to ln ∆0 Λ(~) for T 4i and points (0, qr). S denotes the separatrix Im p = p2V (q). become indistinguishable from the| previous|− one for≃− 1/~ & 3. ± ± With the same substraction, our first proposal (39) gives the constant (ln √π) 0.6 and our second one (44), which is ≃ (i) primitive real periodic orbits r with energy E in the the same as the formula in Landau and Lifshitz [33, sec. 50, problem 3], corresponds to 0 (dotted line). Garg’s formula [27, right region, that is, such that q (E) 6 q(s) 6 q′ (E) and r r eq.1.1] gives the constant (ln pπ/e 0.07) (dash-dotted line). Im p(s) = 0; ≃ (ii) primitive complex periodic orbits c in the central re- gion with purely imaginary p and real q, that is, such ods of the primitive periodic orbits r and c at energy E that qr(E) 6 q(s) 6 qr(E) and Re p(s) = 0; respectively, we have (iii) primitive− real periodic orbits l with energy E in the r 1 η left region. They are obtained from the periodic orbits To(E)= wrTr(E) iwcTc(E) i − Tc(E) (22) by the symmetry S. − − 4 By denoting Tr(E) and Tc(E) the (real, positive) peri- with the winding numbers wr and wc being non-negative 7 o integers. The trajectory may contribute to the deno- in the sum β in formula (19b). When at least one minator (η = +1) or to the numerator (η = 1) of the of the winding number is strictly larger than one, several − right hand side of (9) provided that To = T . For η = +1, staircase time-pathsP can be constructed while keeping re- o is periodic whereas, for η = 1, o is half a symmetric lation (22): they differ one from each other by a different periodic orbit. We will keep the− contributions of all or- partition into steps of the length and/or heights of the bits for which a staircase [t] can be constructed. We can staircase. The corresponding orbits o can be obtained understand from figure 3 that orbits differing by a small one from each other by a continuous smooth sliding of sliding of their initial qi can be obtained by a small mod- the steps of the staircase but during this process one ification of the length or height of the first step. All these cannot avoid an orbit starting at a turning point where contributions are summed up when performing the inte- a bifurcation occurs. gral (17) on one branch β and correspond to one term τ τ′ τ 0 0 0 Tc/2 Tr Tr − τ Tc/2 Tr Tr − τ′ Tc/2 Tr Tr − τ T T T Re p Re q a) Im p b) c) r c −qi −qi −qi l qi qi qi a’) b’) c’) ′ ′ ′ −qi −qi −qi ′ ′ ′ qi qi qi Figure 3: (colour online) For the double well in Fig. 1, at a given energy 0 In the right hand side of (16), the sum involves sev- and wc into integers and by the branch β where its star- eral trajectories differing one from each other by the se- ting point lies. We can therefore express our result in a quence of turning points that are successively encoun- way that can be applied to cases more general than the tered along o. Therefore, to compute the dominant con- double well: the total contribution of the non-zero length tribution of the non-zero length orbits to the numerator paths to the numerator (η = 1) and to the denominator and to the denominator of (9), we will add all the contri- − butions of the topological classes of orbits, each of them uniquely characterised by an ordered sequence of turning points [ρ1,ρ2,... ], in other words by a partition of wr 8 (η =+1) of (9) is separatrix since o µo Tβ dE iSo/~ d Sc(E˜) ( 1) ~ e . (23) = s − √ 2ηiπ dT ˜ ˜ [ρ1,ρ2,... ] r dE Tc(E) ! X X − qr ( E˜) d 4 − s denotes a section of an energy surface in complex phase 2 E˜ + V (q) dq E˜ dE˜ Tc(E˜) − space corresponding to one purely real canonical variable. Z0 q ! q (E) [ρ1,ρ2,... ] is an ordered sequence of (not necessarily dis- 4 dTc(E˜) r = p(q, E) dq< 0 . (26) tinct) turning points that belong to a section s. The sum 2 −Tc(E˜) dE˜ 0 | | concerns all s (different energies may be possible) and Z [ρ ,ρ ,... ] such that we can construct on s, with an ap- 1 2 ˜ propriate choice of [t], a periodic orbit o if η = +1 (a half Therefore Im So reaches its minimum when E is at o its maximum, that is E 0+. The only possible symmetric periodic orbit if η = 1) of period To = T . → The branch β is the one where o starts,− the sequence of equilibrium point contributing to tr Sˆ Uˆ(T ) is the ori- turning points that are successively crossed by o is ex- gin (pf , qf ) = ( pi, qi) = (0, 0); it is also subdominant − actly [ρ1,ρ2,... ]. because H(0, 0) = V (0) = Vmax (which can be seen as ˜ ˜ ˜ In the case of the double-well, for an energy be- the limit of Sc(E)/Tc(E) when E Vmax) is strictly ˜ ˜ ˜ → − low V , a section s for real q has four turning larger than Sc(E)/Tc(E) for E > Vmax. max − points (0, q ) and (0, q′ ). Only the points on s such Assuming that only orbits with real q do contribute ± r ± r that q′ ′ derivative of V ). The differentiation of expressions (33) q (E) r dq with respect to E leads to the asymptotic expansions Tr(E)=2 , (29a) for Tc and Tr. From the first one we can extract the Zqr (E) 2 E V (q) − exponential− sensitivity of E on Im T : qr(E) q dq 2 2 2A ω Im T Tc(E)=4 . (29b) E 2a ω e e . (36) ≃ Z0 2 V (q) E − From relation (28) we can see that 2dTo/dE = q 2wr dTr/dE i dTc/dE is dominated by the last term We have So(E)= wrSr(E) + iSc(E)/2 with if E is small.− ′ q (E) def r dTo i dTc i Sr(E)+ ETr(E)= S˜r(E) = 2 2 E V (q) dq, . (37) q (E) − dE ≃−2 dE ≃ ωE Z r q (30a) Inserting all these asymptotic expansions in the right qr (E) hand side of (32), we get def Sc(E) ETc(E)= S˜c(E) = 4 2 V (q) E dq. − 0 − ~ 2 3 Z q 4 a ω A E/(~ω) S˜c(0)/(2~) (30b) ∆0 e e e− ~∼0 π → r iwrBE(E ~ω)/~ The dominant contributions to tr Uˆ(T ) comes from the e− − . (38) × two stable equilibrium points e = (0, a) for which λe = ± This expression can be turned into the usual jwkb ex- iω. The contribution of (0, 0) is sub-dominant as well as ~ ~ the contribution of any periodic orbit o which necessarily pansion exp a0(E)/ + a1(E) ln + a2(E) + o(1) : As soon as we have condition (6), from expression (36) we turns around about e during Re T then follow an orbit c near S during Im T before coming back to its initial see that E is exponentially small and we obtain a0(0) = − S˜c(0)/2 and a (0) = 1/2. To obtain the correct value point. Together with (6), η = 1, the two stable equili- − 1 brium points give two identical contributions (19a) and of a2, we must proceed to fine-tune the choice of T . A we have criterion is to impose on ∆0 to have a vanishing imag- inary part at any order in ~ consistent with the jwkb iωT/2 tr Uˆ(T ) 2e− (31) expansions used so far. From (32), we will choose E ~∼0 → such that S˜r(E) = ~π, which is exactly the Einstein- which of course could have been deduced directly Brillouin-Keller quantization condition for the ground ~ ~ from E± ~ω/2. Collecting all these results in the right state in one well. This leads to E = ω/2 + o( ). 0 2 ≃ Then Tc = 2 ln ~/(4a ω) /ω +4A/ω + o(1) and hand side of (9), we obtain − ~ ~ 1/2 ω S˜c(~ω/2)/(2~) e ω A S˜c(0)/(2~) 2i~ dTo − ∆ e− =2aω e e− , 0 ~ ∆0 ∼0 √π π ~∼0 π dE → r → r (39) 1 which differs from [27, eqs. (1.1) and (1.4)] by a reason- exp S˜c(E) + (E ~ω/2)Tc(E) × −2~ − able factor √e 1.6. This discrepancy, already noticed in [34, sec. V],≃ which appears in the third order term in iwr exp S˜r(E) (E ~ω/2)Tr(E) ~π . (32) the ~-expansion, comes from the different kind of approx- × ~ − − − imations involved in our approach on the one hand and When E 0+, we have the following asymptotic ex- in Herring’s formula on the other hand. pansions (see→ appendix B) We are also able to obtain a formula for the splitting of the excited states that is consistent with the result 4E √2E 2(2A + 1) given in [27, eq. (B1)]. Using a semiclassical approxima- S˜c(E)= S˜c(0) + ln E + o(E); ˆ ω 2aω − ω tion for the matrix element of Πn, we explain in detail in (33a) appendix C how to obtain ∆n(T ). For one dimensional systems whose energy surface En is made of two branches 2πE 2 2 S˜r(E)= + BE + o(E ) ; (33b) (two Riemann sheets in the complex plane), we can in- ω sert (C8) into (C6) and get one of the main result of this with paper: ~ ˜ ~ a µo+1 iSo(bβ′ ,bβ ,En)/ def ω 1 ∆n(T ) ( 1) e . (40) A = dq ; (34) ~∼0 2T − 2V (q) − a q → [ρ1,ρ2,... ] Z0 − ! X π B def= p5V (3)(a)2 3ω2V (4)(a) (35) To see how formula (40) works in the case of the double- 24ω7 − well potential, we choose the quasi-mode Φ localised | ni 10 on the right torus r at energy En. This torus is made of and where Im T . This regime, that allows to two branches labelled by the sign of p and we can choose select the ground-state→ −∞ doublet only, is included in our a common base point for these two branches, namely approach because the instanton trajectories appear to be b = qr(E). On the symmetric torus l, the two base the limit of c getting closer to the separatrix S whereas ± points will be b = qr(E). Then S˜o(bβ′ ,bβ , En) and the classical real oscillations in the wells shrink to the ± − the index µo do not depend on the choice of the initial equilibrium points. All along this paper we emphasize and final branch. The orbits o that go from r to l must that the phase-space representation is particularly ap- correspond to a T of the form propriate and it is straightforward to recover the usual picture of instantons (for instance q(it) versus it) from 1 our figure 1 d). T = τ + wrTr(E ) i wc + Tc(E ) (41) n − 2 n for non-negative integer (wc, wr) and a fraction of time τ V. DYNAMICAL TUNNELLING FOR THE strictly smaller that Tc(En) that depends on the initial SIMPLE PENDULUM and final conditions (those are not necessarily symme- tric). Then we have The simple pendulum corresponds to V (q)= γ cos q − 1 with γ > 0 and strictly periodic boundary conditions that S˜o(bβ′ ,bβ , En)= wrS˜r(En) + i wc + S˜c(En) (42) identify q = π and q = π. At energy E>γ, the classi- 2 − cal rotation with p = 2(E V (q)) can never switch to − the inverse rotation with p = 2(E V (q)). At the and we take p quantum level, the Schr¨odinger’s− − equation− for the sta- µo = wr + wc +1 . (43) tionary wave-function φ p For the same reason as previously explained the domi- ~2 d2 γ cos q φ(q)= Eφ(q) (45) nant contributions come from those orbits where wc = 0. − 2 dq2 − In order to mimic a real T , we will choose large wind- ing numbers wr such that Re T = wrTr(E ) Im T = n ≫ leads to the Mathieu equation [36], Tc(En)/2. Because of the quantization condition in the right well S˜r(E ) = (n +1/2)2π~, the rapid oscillations y′′(x)+ a 2g cos(2x) y(x)=0 (46) n − exp(i Re So/~) disappear (or inversely if we want to main- ~2 tain ∆n real to first order, we recover the usual quanti- with x = q/2, y(x) = φ(2x), a = 8E/ and g = zation condition). Then we obtain 4γ/~2. The 2π-periodicity of φ forces y to be π- −periodic. The eigenfunctions can be classified accord- ~ 2 S˜c(En)/(2~) ing to the parity operator: Even π-periodic solutions ∆n e− . (44) ~∼0 Tr(E ) exist only for a countably infinite set of characteristic → n values of a denoted by a2n with n =0, 1,... Odd solu- The classical frequency 1/Tr(En) attached to r is es- { } tions correspond to another set, b2n , with n =1, 2 ... sentially of order ω/(2π). ∆ (T ) becomes independent { } n (the a2n+1,b2n+1 correspond to π-antiperiodic solu- of T for large wr: the behaviour of T is mainly gov- tions{ and will be} rejected). The discrete energy spec- erned by wrTr, then the 1/T prefactor in (40) is compen- trum corresponding to even and odd solutions of (45) sated by the increasing number of identical terms in the + ~2 is then En = a2n/8,n = 0, 1 ... and En− = sum since we have seen that the number of topological ~2 { } { b2n/8,n = 1, 2 ... respectively. Any eigenstate φn± classes of orbits increases linearly with 4wr (the factor 4 } | i with energy E± En > γ has its Husimi distribution comes from the two possible initial branches β = and n ≃ ± spread symmetrically between the two half phase-space of the two possible final branches β′ = ; in other words ± positive and negative p, near the lines 2(En V (q)) from the sequences of [ρ1,ρ2,... ] beginning either by qr that define two disconnected tori in the± cylindrical− phase- p or qr′ and ending either by qr or qr′ ). The discrep- space. If we prepare a wave-packet localised on the line ancy between (44) and Garg’s− formula− is just the fac- p = 2(En V (q)), since it is no longer a stationary tor gn given by [27, eq. (B2)] (see also[35, eq. (3.41)]) state, its average− momentum will oscillate between two that tends to 1 when n increases: g = π/e 1.075, p 0 ≃ opposite values, with a tunnelling frequency equal to g1 1.028, ...There is also a ratio of order one, more ∆En/~ where precisely≃ 2/π 0.8, between estimationsp (39) and (44) ≃ ~2 taken for n = 0; the second is slightly better and coin- + p ∆E = E E− = (a b ) (47) cides with the formula given in Landau and Lifshitz [33, n n − n 8 2n − 2n sec. 50, problem 3]. Again, these discrepancies come from the different nature of the approximations that are in- is the splitting between the two quasi-degenerate eigenen- volved. ergies. To compute ∆En, we will use (10) with the ope- ˆ + + Let us end this section by a short comment on the con- rator Πn very much like the exact projector φ φ + | n i h n | nection with the usual instanton theory where Re T =0 φ− φ− : Its matrix element p′ Πˆ p will vanish | n i h n | h | n | i 11 rapidly as soon as p or p′ lie outside the region of the two tori. The main contribution to the semiclassical ex- pansion of tr Sˆ Πˆn Uˆ(T ) will come from classical trajec- tories that connect two symmetric tori. The trace will be semiclassically computed in the momentum basis and we will choose the complex time path to maintain p real. To construct one trajectory at energy E>γ connecting the two tori requires to have purely imaginary q when- def ever pr dp dt 1 = γ sh q i ; (49a) ds ± 2 ds dq dt 2 = p i (49b) ds 1 ds and are associated with a real time dynamics governed by the Hamiltonian H˜ (p , q )= p2/2 γ ch q . (50) Figure 4: (colour online) With the choice of a complex- 1 2 1 ∓ 2 time path given in 3) (one-step path), the contributions to tr Sˆ Πˆn Uˆ (T ) come from one half symmetric periodic or- with “ ” corresponding to case (i) and “+” correspond- − bits o (b) that are a composite of 1) (repetitions of) a periodic ing to case (ii). Then, instantons correspond to trajecto- r r def orbit of period T in the phase-space plane (Re q, Re p) at ries evolving in the transformed potential V˜ (q) = γ ch q energy E>γ (recall that the planes q = π are identified); ∓ ± rather than the usual inverted potential V (q). We 2) one half of a periodic orbit c of period Tc in the phase-space will choose a one step complex time path− as in Fig. 3 plane (Im q, Re p) at energy E. and we will represent the orbits in a three dimensional space (Re q, Re p, Im q) and the connection between the allowed and the forbidden trajectories occurs on the axes (Re q = 0, Im q = 0) and (Re q = π, Im q = 0) (Fig. 4a). Trajectories in family (i) escape± from the un- imaginary part of the action, expression (13) becomes stable point at Re p = 0, Im q = 0 without coming back to the plane Re p = 0, Re q = 0. Only orbits in family (ii) can be used to produce periodic orbits. A typical pe- s f dq riodic orbit o connecting two symmetric rotations of the S = p (s) 1 (s)ds E Re T 1 ds − pendulum is given in Fig.4b). We will choose Im T to be Zsi sf precisely the half period of the periodic orbit c of fam- dq2 + i p (s) (s)ds + E( Im T ) . (51) ily (ii) at energy E and Re T to be (an integer multiple 1 n si ds − of) the period of rotation of the pendulum at the same Z energy. This choice exhibits the dominant contribution to tr Sˆ Πˆn Uˆ(T ) . We can reproduce the same reasoning that led to (44) with the rˆole of p and q being exchanged. Then Now Tr = dS˜r/dE is the typical frequency on the torus at energy En. Keeping only the contributions that provide a positive So(E)= S˜o(E) ETo (52) − 12 with To = T and n =1 S˜o(E)= wrS˜r(E) + iS˜c(E)/2 a) π =2wr 2(E + γ cos q) dq n =9 Z0 argch(pE/γ) + 2i 2(E γ ch q) dq . (53) 0 − Z p All these expressions can be written in terms of the com- plete elliptic integrals [37] (defined for u < 1) | | π/2 dx (u) def= ; (54a) K 2 2 Z0 1 u sin x π/2 − 1/~ (u) def= p1 u2 sin2 x dx . (54b) E 0 − Z p Namely, b) 2γ S˜r(E)=4 2(E + γ) ; (55a) E E + γ r p E γ E γ S˜c(E)=8 2(E + γ) − − . "K sE + γ ! −E sE + γ !# p (55b) Then we obtain ~ 2 S˜c(En)/(2~) ∆n e− (56) ∼ Tr(En) ~ with 1/ dS˜r 2√2 2γ Tr(E)= = . (57) dE √E + γ K E + γ r The energies of the highly excited states are approxi- Figure 5: For the pendulum with γ = 1, we have plotted mately given by the free rotations: E n2~2/2 γ. in a) the exact value of ln(∆En) (see eq. (47)) versus 1/~ n ≃ ≫ Then Tr(En) 2π/(n~). The asymptotic expansion for n = 1,... 9 (black lines). The dots are given by the ˜ ≃ semiclassical approximation ln(∆n) (see eq. (58b)) up to the of Sc(E) for large E leads to + maximum value of 1/~ (vertical dotted lines) where En be- 2 come lower that the separatrix energy γ. The semiclassical ~ √2E ~ n 2(ln(γ/En)+2 3ln2)√2En/ limit is obtained when the states become more and more lo- ∆n ~ e − ; (58a) ∼ π calised in the region of phase space corresponding to rota- 1 e 4n γ 2n 2~2 ~2 . (58b) tions: for a fixed n this requires to increase En n /2 that 4n 1 ~2 ~ ≃ ∼ πn − 2 is to increase ; or for a fixed energy E >γ, we must in- crease n √2E/~. To control the validity of the prefactors, ∝ The last expression corresponds exactly to equa- we can check in b) that ln(∆En/∆n) approaches zero in the tion (3.44) of [35] obtained with standard uniform semi- semiclassical limit. classical analysis. We see on figure 5 that (56) is a very good approximation even when the energies En get close to γ the energy of the separatrix. deeper central well (Fig. 6 a). The minimum of the right and left wells is fixed at zero, the minimum of the central well is V (0) = Vmin < 0 and the local maximum between VI. RESONANT TUNNELLING AND the wells is denoted Vmax. When 0 inant contributions, when o remains as shortly as possible with complex p. Then, for such orbits to exist, we must choose T of the form 1 T = τ + wrTr(E)+ wm + Tm(E) iTc(E) (59) 2 − involving the periods of the primitive orbits and the corresponding winding numbers wr, wm which are non- negative integers. τ denotes a positive fraction of time smaller than Tr(E). The base points for the two branches β = defined by Re p ≷ 0 on r coincide with ± ˜ ′ the turning point b = qr′ , the action So(bβ ,bβ , En) and ± the index µo are independent of the choice of the branch where o starts: 1 S˜o(b ′ ,b , E )= wrS˜r(E)+ wm + S˜m(E)+iS˜c(E) . β β n 2 (60) µo = wr + wm +3 . (61) Explicitly, we have ′′ q (E) def r S˜r(E) = 2 2 E V (q) dq ; (62a) q′ (E) − Z r q qr (E) def S˜m(E) = 4 2 E V (q) dq ; (62b) 0 − Z ′ q (E)q def r S˜c(E) = 2 2 V (q) E dq ; (62c) − Zqr (E) q Figure 6: (colour online) To describe resonant tunnelling and the corresponding periods Tr, Tm, Tc are obtained between two symmetric wells centered at a and separated by deriving with respect to E. by a third central well (a), we use orbits made± of five primitive trajectories with a time path given in (d). We still maintain As we have seen, formula (40) will provide an approx- the position q to be real and can use a three dimensional imation of the exact splitting ∆En that becomes better, the better the condition Re T (E ) Im T (E ) is sat- section of phase-space (see text). Here we have chosen V n ≫ n given by (65) with a = 7/4 and b = 1/2. isfied. Not only this condition render the precise value of τ irrelevant, it also requires large wm and/or wr, espe- cially if we work with E & 0 for which Tc(E ) Tm(E) n n ≫ and Tc(E ) Tr(E). For a given pair of wr and wm, energy 0 obtain S˜c(En)/~ ~ e− ln |∆En/∆n| ∆n(T ) . (66) | |~ ∼0 Tr sin π(mνr rνm) 0 0 → − Fig. 7 shows that this latter expression provides a good approximation for ∆En even in the immediate neigh- -5 -5 bourhood of a resonance where estimation (10) is not justified any more. If we had continued working with a finite K, the sum (63) would have involved a finite num- -10 -10 ber of terms and the singularities due to the vanishing denominators in (66) would have been smoothed down -18 (inset in Fig. 7). In other words, for a fixed T , rotating -15 -15 down T in the lower half plane Im T < 0, destroys| | very -20 quickly (θ 0.2) the large resonant fluctuations of tun- K = 100 nelling. This≃ effect has already been shown in the case K = 30 -20-22 -20 of a kicked system [28]. -24 -25K =5 -25 VII. ESCAPE RATES -26 6.2 6.4 6.6 6.8 7 So far we have focused our analysis of tunnelling in -30 -30 bounded systems only, but the philosophy we presented 2 3 4 5 6 7 here can be extended to more general situations. For ins- 1/~ tance, let us show how we can compute the escape rate from a metastable state localised in a confining poten- tial V whose shape has the form given in Fig. 8 a). The Figure 7: (colour online) For V given by (65) with a = 7/4 potential has a local minimum at q = 0 (say Vmin = 0) and b = 1/2, we plot ln ∆En (thick black line) for the low- | | ± and an energy barrier for 0 2 ~ Tm m Γ Im eiEnT/ tr Πˆ Uˆ(T ) (69) (64) n ≃−Im T n Tr ≃ r which allows an explicit semiclassical expansion in terms with m and r being coprimes positive integers. For the of classical solutions with a complex time path. The dom- polynomial potential inant contributions will be provided by periodic orbits in time T with real q starting on the torus at energy En, V (q) = (q2 a2)2(q2 b2) (65) − − then, while idt/ds > 0, going forth outside the well be- fore coming back to its initial starting point (Fig. 8 c, with a>b> 0, the argument presented in [38] can be d)). Then, up to a dimensionless factor fn of order one, generalised to show that (64) is actually exact with r =1 we get and m = 2 for any energy 0 with the same weight and therefore the escape rate should twice as large. In the case of an island with sharp bound- aries around qmax, S˜c(E) which is the area enclosed by the primitive orbit c (Fig. 8 d) is mainly given by the portion of c where we can keep an harmonic approxima- tion for the potential: V (q) ω2q2/2, q (E) √2E/ω, ∼ r ≃ qr′ (E) A/(πω) where A is the area of the island in the real≃ phase space. Then, with a def=ωA/(2πE), we have p 2E S˜c(E) √a√a 1 ln √a + √a 1 . (72) ≃ ω − − − Inserting this expression in (69) with En (n +1/2)~ω, we exactly recover the expression (5) used≃ in [39] with an elegant and simple interpretation. The chaotic sea that surrounds the integrable island in the mixed system con- sidered by B¨acker et al. acts as a sharp effective potential barrier as the one draw in picture 8; our complex trajec- tory that allows to escape from the regular region has its main features governed by the integrable (and even har- monic) approximation of the dynamics about the island, following precisely the general philosophy of [39] and [40]. This computation of the “direct” tunnelling (by opposi- tion to resonant tunnelling where the model of a pure quadratic kinetic energy fails) can also be reproduced within the standard one-dimensional JWKB theory used for computing transmission coefficients. Here again, we can check easily that the tradition- nal instanton method is included in our approach : The regime T 0 selects the instanton solution and pro- vides theℜ escape→ rate from the equilibrium point as given by equation (2.47) of [2, chap. 7]. Figure 8: (colour online) The escape rate at energy E from an island created by a confining potential V whose shape is shown in a) can be semiclassically computed in terms of com- plex classical orbits where one canonical coordinate, say q, is VIII. CONCLUSIONS kept real. In phase-space, this trajectory appears to be a con- catenation of two types of curves that join on the (Re q)-axis: The explicit semiclassical expansions of trace formu- 1) a trajectory that lies in the phase-space plane (Re q, Re p) lae for tunnelling splittings (or escape rates) in terms with real variations of t at energy E with the potential V and 2) a trajectory that lies in the phase-space plane (Re q, Im p) of classical orbits constructed with complex-time paths with imaginary variations of t at energy E with the poten- provide an interesting alternative approach to Herring tial V shown in b). In c) a family of constant− energy curves formulae essentially because they do not require to an- is shown− (horizontal green for the first type, vertical red for alytically continue the wave functions in the complex the second type). In d) for a given energy, we show how three plane. In multidimensional tunnelling and/or for a non- curves l, c, r glue together at the turning points. autonomous Hamiltonian system, the generic lack of con- stants of motion isolates the stable islands (if any) from each other by chaotic seas. The analytic continuation with of the kam tori that build the islands is prevented by ′ the existence of natural boundaries (see for instance the q (E) def r recent discussion in [30, part II, appendix B] and ref- S˜c(E) = 2 2 V (q) E dq (71) − erences therein). The approach we have presented here Zqr (E) q seems to circumvent these difficulties but, of course, the where 0 < qr(E) < qr′ (E) are the two right turning points problem of how to select, with an appropriate [t], the rel- at energy E. Here we have supposed that the action be- evant trajectories among an a priori exponentially grow- tween the two left turning points is larger than (71) and ing number of classical complex solutions remains open. gives a subdominant contribution. If the potential is sym- We expect the tunnelling splittings between two states metric, two symmetric complex orbits would contribute at energy around E localised in two symmetric islands to 16 be approximated by the expansion of the form Given a [t], for any classical phase-space path o, we can consider the final coordinates (pf , qf ) at time t(sf )= T ~ N M ′ ′ as smooth functions of the initial coordinates (p , q ) at f i P wi[S˜i(En)/~ π] P w [S˜ (En)/~] i i e i=1 − e− j=1 j j T time t(si) = 0. The monodromy matrix Mo is defined [ρ ,ρ ,... ] 1X2 as the differential of these functions: to first order, an (73) initial small perturbation implies the final perturbation where the sum runs over all possible sequences of turning δpf Mo Mo δpi = ,11 ,12 . (A1) points [ρ1,ρ2,... ] at energy E such that we can choose δqf Mo,21 Mo,22 δqi a complex-time path that leads to a trajectory, made of primitive orbits, that connects in time T the two (real) The sub-matrices can be expressed from the sec- tori. The winding wi and the actions Si (resp. wj′ ond derivatives of So(qf , qi; T ) by differentiating equa- ˜ { } { } { } and Sj′ ) refer to the primitive orbits obtained when the tions (18): variations{ } dt/ds are purely real (resp. purely imaginary). 1 For dimensions larger that one, the dimensionless prefac- 2 2 − Mo,11 = ∂q q So ∂q q So ; (A2a) tor f may appear as a power law in ~. Here, inspired by − f f f i 1 the study in section VI, we can qualitatively see how the 2 2 2 − 2 Mo,12 = ∂q q So ∂q q So ∂q q So ∂q q So ; (A2b) constructive interferences between repeated paths emerge f i − f f f i i i 1 in a speckle-like forest because of the presence of resonan- 2 − Mo,21 = ∂ So ; (A2c) ces. As shown in [28], a progressive complex rotation of − qf qi 1 time provides a natural way to select the main resonance 2 − 2 Mo = ∂ So ∂ So ; (A2d) effects. If we want to expand the splittings (or the es- ,22 − qf qi qiqi cape rates) according to elementary process as proposed 2 provided that ∂ So is invertible. Equations (A2a,c,d) in [40], our approach offers a promising tool to interpret qf qi and compute semiclassically all of the ingredients of such can be inverted in an expansion. 2 1 ∂ So = (Mo )− ; (A3a) qf qi − ,21 2 1 o o o ∂qiqi S = (M ,21)− M ,22 ; (A3b) Acknowledgments 2 1 o o o ∂qf qf S = M ,11(M ,21)− ; (A3c) We have received a lot of benefits from discussions with while (A2b) provides Olivier Brodier, Dominique Delande, Akira Shudo, De- 1 1 nis Ullmo and Jean Zinn-Justin, with special thanks to Mo,12 = (Mo,21)− + Mo,11(Mo,21)− Mo,22 (A4) Stephen Creagh, who has had a good intuition about − these issues for a long time and has shared his insights which is nothing but the expression that det Mo = 1 once with us. We acknowledge Stam Nicolis for his careful we use the identity reading of the manuscript and Olivier Thibault for his A B 1 efficient skills in terms of computer maintenance. One of det = det(CAC− D CB) (A5) CD − us (A.M.) is grateful to the Laboratoire Kastler Brossel for its hospitality. where (A,B,C,D) are square matrices of the same size and C is invertible. Appendix A Coming back to the oscillating integral (17), if the in- tegration path can be deformed in order to pass through an isolated critical point qc of q So(ηq,q,T ), we will In this appendix, we derive the dominant contribu- have a contribution of the form 7→ tions (19) to the traces involved in (9). Both cases ˆ tr Uˆ(T ) and tr S Uˆ(T ) will be treated simultaneously ∂2So by defining the sign η to be +1 in the first case and 1 det ∂qi∂qf − s o ~ in the second case. The semiclassical arguments under- eiS / (A6) pinning the derivation are relatively standard and may ∂2 So(ηq,q; T ) det be found in one way or another in the literature. For v u − ∂q∂q ! instance, the contribution (A13) for η = 1 can be found u in [41, eq. (2.12)] within a more restricted context (see t also [42, chap. 8]). Nevertheless we found it useful to pro- where all the functions are evaluated at (ηqc, qc,T ). vide all the steps in the precise context of this work, not Physically qc is interpreted as the initial position of the o only to render the presentation self-contained, but also phase space paths such that (pf , qf ) = (ηpi, ηqi). With because we are working in the time domain with general the help of (A3), the prefactor of the exponential can be 1 1/2 Hamiltonians that have not necessarily the form (1). written as det Mo +Mo Mo (Mo )− 2η − ,22 ,21 ,11 ,21 − 17 up to a global sign. Using (A5) again, the contribu- we may encounter several possible branches pβ(q, E) for tion (A6) can be written as the graph of these functions (the two possible signs of a square root when the Hamiltonian has the form (1)) νo ( 1) iSo/~ which become singular but pairwise connect smoothly − e (A7) ( η)d det(1 ηMo) at the turning points, defined by ∂pH = 0. A relation − − between E, T , qf and qi can be obtained by integrat- where ( 1)νo pfixes the sign of the square root and d de- ing sf (dt/ds) ds and using (14b): si motes the− number of degrees of freedom. Recall that in R sf the definition of the path-integrals (12), a time-slice τ is def dq/ds T = To(q , q ; E) = ds . always implicit. Let us keep for a moment an explicit f i si ∂ H p q(s), E , q(s) discretization, for instance with T being an integer mul- Z p tiple of τ, o referring to a discrete set of points, Hamil- (A10) ton’s equations (14) being discretized into a phase-space Everywhere but at the turning points, the value of s map, path integral (13) being turned into a discrete (Rie- dictates the choice of the branch used for the inte- mann) sum, etc. Then, the contribution of each of the grand. This relation implicitly defines Eo(qf , qi; T ). points of a given o, such that (pf , qf ) = (ηpi, ηqi), is the The usual expression for the derivative of implicit same and remains given by (A7); but this is correct only functions leads to the relations: ∂E p =1/∂pH and ~ ∂ Eo = ∂ To/∂ To = ∂ Eo/∂ H(p , q ). If we dif- if, for a given τ, is small enough, because when per- qi − qi E T p i i forming the steepest-descent method on (17), one must ferentiate (18b) with respect to qi when pf is given be able to split the oscillating integral into separate con- by p qf , Eo(qf , qi,T ) , we obtain tributions coming from two distinct points of o. In the 2 continuous time limit, i.e. when the limit τ 0 is taken ∂ So 1 1 ∂Eo → = . before the semiclassical limit ~ 0 (see [??]), the or- ∂qi∂qf ∂pH(pf , qf ) ∂pH(pi, qi) ∂T (qf ,qi;T ) (qf ,qi;T ) o → bits of non-zero length appear as a one-dimensional (A11) continuum none of whose points can be considered sep- With property (3), we get ∂pH(ηp,ηq) = η∂ pH(p, q) arately anymore. The only isolated critical points qc are and then, by differentiating (A10), we have given by the position of some equilibrium points. Under ∂q To(ηq,q,E) = 0. Therefore, the energy Eo of the symmetric condition (qi,pi) = ( pi, qi), only the the path o depends only on T , not on its starting − − origin must be examined. Linearising the Hamiltonian point q. The square root of ∂T Eo(ηq,q,T ) = dEo/dT flow about a non-degenerated fixed point e = (pe, qe) can be got out from the integral in (A9). When adding leads to a monodromy matrix whose eigenvalues can be the contribution of each path whose starting point lie on collected by pairs exp( λe,αT ) where λe,α α=1,...d are the branch β, we obtain the Lyapunov exponents.± Then (A7) becomes{ } def dq νe iH(pe,qe)T/~ Tβ(E) = . (A12) ( 1) e− − (A8) Z ∂pH pβ q, E , q d λe,αT/2 λe,αT/2 (e ηe− ) α=1 − This is not exactly the right hand side of (A10) because Q with a possible adjustment of the sign. For a generic the domain of integration in (A12) is the domain of the choice of T , the denominator does not vanish. branch where the starting point of o lives. Each branch β For a non-zero length path o, the critical q’s are de- is delimited by two turning points and Tβ is the time generate along the trajectory; for a system with several spent to go from one point to the other. degrees of freedom, one must treat separately the (gaus- The integral (A9) involves all the possible starting sian) integrals on the transverse coordinates q along ⊥ points for a trajectory o and therefore we must add all which So(ηq,q,T ) varies (quadratically) from the longi- the branches that patchwork smoothly in phase-space to tudinal coordinates q along which So(ηq,q,T ) is cons- form the geometrical set of points crossed by o. Referring tant. Of course, thek dimensions of q and q depend ⊥ k to the purely geometrical quantities (i.e. independent of crucially on the presence of kam tori. However, multi- the choice of the parametrization), we have the contribu- dimensional tunnelling is beyond the scope of this paper tion and, the quantitative studies presented here concern one- dimensional systems only. The contribution to the trace T β β dEo ~ of such a path is then, up to a global sign, ( 1)µo eiSo/ (A13) − √P2ηiπ~ dT − r iSo/~ e ∂2So dq . (A9) only if To = T ; for η = +1 the path o is a periodic orbit √ 2iπ~ ∂qi∂qf o Z s (ηq,q;T ) and, for η = 1, the path is half a symmetric perio- − − dic orbit (the whole periodic orbit being of period equal The conservation of energy along o, H(p, q)= E, implic- to 2T ). The sum concerns all the geometrical branches β itly defines a function p(q, E) in the neighbourhood of any crossed by o (even if o passes several times by the same point where ∂ H = 0. Globally along the trajectory o, points, each branch is only counted once). As before, the p 6 18 conversion of a product of two square roots of complex The function L(s,ǫ) is not continuous but we can extract numbers to the square root of the product may introduce the discontinuous part from a sign that can be absorbed in the definition of µo; the 1 exact computation of the index µo is difficult but since − 2 4 ( 1)n2V (n)(a) sn ǫn it may change at the bifurcation points only, where the L(s,ǫ)= 1+ − − √ 2 2 n! s2 ǫ2 semiclassical approximation fails, it is sufficient to know ω s ǫ > − nX3 − that it depends on the nature and the number of the (B6) o turning points encountered on . Therefore it is an addi- by expanding the last factor: tive quantity when several primitive orbits are repeated or concatenated together. 4 2V (3)(a) s3 ǫ3 L(s,ǫ)= + − + M(s,ǫ) ω√s2 ǫ2 3ω3 (s2 ǫ2)3/2 − − (B7) Appendix B where now M(s,ǫ) is a continuous function of its two variables. Then, a standard theorem in analysis as- sures that ǫ a M(s,ǫ) ds is continuous and its limit In this appendix we explain how to obtain the asymp- 7→ ǫ + when ǫ 0 is totic expansions (33) as E 0 . → R ˜ → ˜ ˜ First consider Sr and split it in two parts S+ + S a a 4 2V (3)(a) where − M(s, 0) ds = L(s, 0) ds − ωs − 3ω3 Z0 Z0 ′ (3) qr (E) 4A 2aV (a) S˜ def= 2 2 E V (q) dq ; (B1a) = (B8) + − ω − 3ω3 Za a q S˜ def= 2 2 E V (q) dq . (B1b) with A given by (34). The two other integrals obtained − − by inserting (B7) in the right hand side of (B4) can be Zqr (E) q computed exactly and expanded as ǫ 0+ up to or- → def der o(1). Then, inserting (B3), we obtain Setting ǫ = a qr(E), rewrite S˜ as − − 1 dS˜c 4 √2E 4A = ln + o(1) . (B9) S˜ =2√2ǫ V (a ǫ) V (a sǫ) ds, (B2) dE ω 2aω − ω − 0 − − − ! Z p Now expand the integrand as a power series in ǫ up to the Its integration leads directly to (33a). fourth order, compute the integrals that appear in each coefficient and insert the expansion of ǫ in E obtained from the implicit equation V (a ǫ)= E: Appendix C − def + √ √2 V (3)(a) The quasi-mode Φn =( φn + φn− )/ 2 is localised √ | i + | i | i ǫ = E + 4 E on one torus at energy En En−. Standard jwkb tech- ω 3ω niques [43, 44] provide a semiclassical≃ approximation to 2 √2 3ω2V (4)(a) 5 V (3)(a) its wave function: − E3/2 + O(E2) . − 36ω7 (B3) s.c. 1 An,β Φn (q)= β Tβ(En) β ∂pH pβ(q, En), q Proceed in an analogous way for the computation of the X q q first three terms of the asymptotic expansion in √E for P q exp i pβ(x, En) dx/~ (C1) S˜+. When summing S+ and S , expression (33b) is ob- × bβ ! tained with (35). − Z ˜ The expansion of Sc is more subtle since it is not dif- (β labels the possible several branches of the torus, An,β ferentiable at E = 0. Its derivative is given by are dimensionless coefficients of unit modulus, bβ is a base point of the branch β and Tβ(En) the character- a dS˜c istic time (A12) spent on the branch β). Within the = L(s,ǫ) ds (B4) semiclassical approximation, it is, therefore, consistent to dE − ǫ Z construct Πn(q′, q) by substituting (C1) in the matrix el- + + def ements of the projector operator φn φn + φn− φn− = where we denote ǫ = a qr(E) and define | i h | | i h | − Φ Φ + Sˆ Φ Φ Sˆ. From the integral (20) | ni h n| | ni h n| def 2√2 s.c. s.c. L(s,ǫ) = . (B5) 2 dq dq′ Φ (q) Φ (q′) ∗ G(ηq′, q; T ) , (C2) V (a s) V (a ǫ) n n − − − Z p 19 when we insert the semiclassical expressions (16), we ob- that depend only on the geometry of the orbit, not on the tain a sum of integrals of the form number of times the considered branch may be repeated as s goes from si to sf . As discussed in the case of the ′ o i q i q double well, if there exist different topological classes of , p ′ (x, E ) dx ~ pβ(x, En) dx ~ β n each of them being characterised by an ordered sequence b − b ′ e Z β e Z β of turning points [ρ1,ρ2,... ], we must add such contri- dq dq′ iEnT/~ × butions. Then, using directly tr ΠˆnUˆ(T ) 2e− , ∂ H p (q, E ), q ∂ H p ′ (ηq , E ), q Z p β n p β ′ n ′ we have proven that ≃ q q 2 o ′ ∂ S iSo(ηq ,q;T )/~ ~ o ˜o ′ ~ e . (C3) µ ′ iS (bβ ,bβ ,En)/ ∂qi∂qf ∆n(T ) ( 1) Aβ,β (En,T )e s (ηq′ ,q;T ) ~∼0 T − → [ρ1,ρ2,... ] X The stationary conditions (C6) where the sum runs over all the sequences of turning s pβ(q, En) = ∂qi So(ηq′, q; T ); (C4a) points on the section at energy En where one canonical − variable is maintained real. There must exist for such ηp ′ (q′, E ) = ∂ So(ηq′, q; T ) (C4b) β n qf a sequence, one half symmetric orbit o starting on the branch β of the torus at energy E , crossing successively select the classical trajectories o with energy En that go n all the sequences [ρ ,ρ ,... ] and ending on the branch β from (pi, qi) = (pβ(q, En), q) at t(si) = 0 to (pf , qf ) = 1 2 ′ ′ (ηp ′ (q , E ), ηq ) at time t(s ) = T . Then the value of at time T . The dimensionless coefficients Aβ,β (En,T ) β ′ n ′ f ~ the exponent have a -independent modulus of order one and depend only on the geometrical properties of the branches. If So,β,β′ (En,T )= EnT + S˜o(bβ′ ,bβ , En) (C5) some parts of the trajectory are repeated, their repeti- − ′ tion numbers do not appear in Aβ,β (En,T ) but only in depends only on the branches where the starting and the cumulative quantities: the index µo and the action ending points lie and not on the precise location of these S˜o(bβ′ ,bβ , En) given by points on the branches. Since q and q′ correspond to the same torus, such a trajectory must connect the two sf dq ˜o ′ symmetric tori for η = 1. At a given En and T , for S (bβ ,bβ , En)= p(s) ds − si ds a fixed q = qi on the branch β, qf and β′ are uniquely Z q(s ) q(s ) given and we can make the stationary phase approxi- i f + pβ (x, En) dx pβ′ (x, En) dx . (C7) mation for the integral on q′. Then if we insert (A10) − ′ Zbβ Zbβ at energy En into (C4) and differentiate it with respect to q or q′, we obtain some identities that, with (A11), In the case of two branches, the computation of the coef- allow us to simplify the combination of the prefactors ficient can be done exactly using the appropriate choice and the remaining integral in q turns out to be precisely of phase conventions for the base points bβ and An,β; we of the form of the right hand side of (A12). 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