Some tunneling phenomena are described, in the semiclassical approximation,
by unstable complex trajectories. We develop a systematic procedure to
stabilize the trajectories and to calculate the tunneling probability,
including both the suppression exponent and prefactor. We find that the
instability of tunneling solutions modifies the power-law dependence of the
prefactor on h as compared to the case of stable solutions.Comment: Journal version; 4 pages, 2 figure

A. G. Panin

D. G. Levkov

F. Bezrukov

S. M. Sibiryakov

English

arXiv

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Unstable Semiclassical Trajectories in Tunneling

Some tunneling phenomena are described, in the semiclassical approximation, by unstable complex trajectories. We develop a systematic procedure to stabilize the trajectories and to calculate the tunneling probability, including both the suppression exponent and prefactor. We find that the instability of tunneling solutions modifies the power-law dependence of the prefactor on h as compared to the case of stable solutions.Some tunneling phenomena are described, in the semiclassical approximation, by unstable complex trajectories. We develop a systematic procedure to stabilize the trajectories and to calculate the tunneling probability, including both the suppression exponent and prefactor. We find that the instability of tunneling solutions modifies the power-law dependence of the prefactor on h as compared to the case of stable solutions

Levkov, D.G.

Panin, A.G.

Sibiryakov, S.M.

CERN Document Server

arXiv:0707.0433v2  [quant-ph]  6 Nov 2007CERN-PH-TH/2007-109Unstable Semiclassical Trajectories in TunnelingD.G. Levkov,1 A.G. Panin,1, 2 and S.M. Sibiryakov3,11Institute for Nuclear Research of the Russian Academy of Sciences,60th October Anniversary prospect 7a, Moscow 117312, Russia2Moscow Institute of Physics and Technology, Institutskii per. 9, Dolgoprudny 141700, Moscow Region, Russia3Theory Group, Physics Department, CERN, CH-1211 Geneva 23, SwitzerlandSome tunneling phenomena are described, in the semiclassical approximation, by unstable com-plex trajectories. We develop a systematic procedure to stabilize the trajectories and to calculatethe tunneling probability, including both the suppression exponent and prefactor. We find thatthe instability of tunneling solutions modifies the power–law dependence of the prefactor on ~ ascompared to the case of stable solutions.Tunneling in systems with many degrees of freedomhas been a subject of continuous theoretical research forthe last decades [1]. The interest is heated up by therecent experimental observations [2–4] of non–trivial dy-namical properties of multidimensional tunneling. In thesemiclassical framework tunneling is described as motionof the system along a complex trajectory – solution tothe classical equations of motion analytically continuedto the complex values of coordinates and/or time [5, 6].Given the complex trajectory, one calculates the tunnel-ing probability at small values of the Planck constant ~,P = Ae−F/~ , (1)where F and A are the suppression exponent and pref-actor, respectively.Recently a new, intrinsically multidimensional, mech-anism of tunneling has been discovered [7, 8]. It differsqualitatively from the well-studied case of direct tunnel-ing, where complex trajectories connect the in- and out-regions of the phase space. The new mechanism gener-ically occurs in the situation when the total energy ofthe system exceeds the height of the potential barrierseparating the in- and out- states but, still, the processremains exponentially suppressed (dynamical tunneling).The complex trajectories in the new mechanism end upon a real unstable periodic orbit lying on the bound-ary between the in- and out- regions. We call this or-bit “sphaleron”; its instability implies that the tunnel-ing trajectories are also unstable. The above behaviorof complex trajectories leads to the physical picture oftunneling as a two-step process [7, 9]: formation of thesphaleron and its decay into the out-region. The latterstep is not described by the tunneling trajectory; on theother hand, it does not involve exponential suppression,as the decay of the sphaleron proceeds classically. Thistunneling mechanism is generic and has been found inseveral quantum mechanical [10, 11] and field theoreti-cal [9] models. It is natural to call the new mechanism“sphaleron–driven” tunneling [12].Tunneling via unstable semiclassical solutions raises anumber of issues. First, search for unstable trajectoriesis problematic from the numerical point of view. Second,even if one finds the tunneling trajectory which tends tothe sphaleron as t→ +∞, a problem remains to describesemiclassically the subsequent decay of the sphaleron or-bit. Yet another issue is the calculation of the prefactorA. In the case of direct tunneling this calculation involvesthe analysis of linear perturbations around the tunnel-ing trajectory. This procedure is not applicable in thesphaleron–driven case, when the perturbations destroythe tunneling solution completely.In this Letter we systematically develop a generalmethod to solve the above problems. The idea is to in-troduce a constraint into the path integral for the tunnel-ing amplitude. This modifies the equations of motion insuch a way that tunneling trajectories are pushed awayfrom the sphaleron. The modification is governed by aregularization parameter ǫ. At ǫ > 0 the semiclassical so-lutions interpolate between the in– and out–regions andare stable. They describe the whole two-stage processof sphaleron-driven tunneling. Integration over the con-straint corresponds to taking the limit ǫ→ +0. The orig-inal unstable trajectory is recovered from the regularizedsolutions in this limit. We find expressions for the sup-pression exponent and prefactor of the sphaleron-driventunneling probability in a closed form. Our method isbased on the ideas of [7].Our analysis reveals, among other things, one universalfeature: the prefactor A in the probability of tunnelingvia unstable trajectories gets suppressed by an additionalfactor ~1/2 as compared to the case of direct tunneling.This, in particular, implies non-trivial properties of thetransition between the direct and sphaleron–driven tun-neling regimes.Remarkably, the method of ǫ-regularization can beused to deform real solutions corresponding to purelyclassical motion into trajectories describing tunneling[7, 10]. This makes the method efficient for finding andclassifying complex solutions in the case of chaotic tun-neling [10]. We stress, however, that the phenomenon ofsphaleron–driven tunneling is unrelated to chaos and wasobserved both in chaotic and regular systems.While our approach is completely general, for concret-ness, we illustrate it using a two–dimensional model with2the HamiltonianH = (p2x + p2y + ω2y2)/2 + exp[−(x+ y)2/2] . (2)The model describes the motion of a particle in a po-tential valley extended along the x-axis with quadraticconfining potential in the y-direction. The valley is in-tersected at an angle by the potential barrier which in-troduces non-linear coupling between the degrees of free-dom. The process we are interested in is a penetrationthrough the barrier of the particle which comes from theleft in a fixed initial quantum state |E,Ey〉. The latteris characterized by the total energy E and the energy ofoscillations in the y-direction, Ey = ~ω(n+ 1/2), wheren is the occupation number of y–oscillator. Note that wekeep Ey fixed in the semiclassical limit ~ → 0, so that ngrows to infinity. It is shown in [7] that, for given Ey,transmission through the barrier is a tunneling processfor total energies E < Eb(Ey), while at E > Eb(Ey) thetransmission proceeds classically. The values of Eb areconsiderably higher than the height of the potential bar-rier, Eb(Ey) > Vmax = 1. The mechanism of transmis-sion changes from direct tunneling to sphaleron–driventunneling when the total energy exceeds a certain criti-cal value Ec(Ey), where Vmax < Ec(Ey) < Eb(Ey).We start by reviewing the derivation of the formula(1) in the ordinary regime of direct tunneling. One con-siders the amplitude of transition from the state |E,Ey〉at the initial time moment t = ti to the state |xf 〉 withdefinite coordinates beyond the barrier at the final mo-ment t = tf . Using the propagator in the coordinaterepresentation, one writes,A =∫dxi〈xf |e−iH∆t/~|xi〉〈xi|E,Ey〉 , (3)where ∆t ≡ tf − ti. The propagator is given by thesemiclassical Van Vleck formula,〈xf |e−iH∆t/~|xi〉 = eiScl(xi,xf )/~2πi~[det∂2Scl∂xi∂xf]1/2, (4)where Scl(xi,xf ) is the action evaluated on the classicalsolution going from xi at t = ti to xf at t = tf . It willbe important for us that (4) can be derived [13] from thepath integral〈xf | e−iH∆t/~|xi〉 =∫x(tf )=xfx(ti)=xi[dx(t)] eiS[x(t)]/~ . (5)We take the wave function of the initial state in the form〈xi|E,Ey〉 = ψ(xi)Ψ(yi), where ψ(xi) is the plane wavewith the unit flux normalization, and Ψ(yi) is the semi-classical expression for the oscillator eigenfunction. Eval-uating the integral (3) in the saddle–point approximationone obtains,A =√ω2πD1ei(Scl+B)/~+iπ/4 , (6)whereB = x˙ixi +∫ yi√2Ey/ωdy′√2Ey − ω2y′2 ,D1 = x˙iy˙i det∂2(Scl +B)∂x2i[det∂2Scl∂xi∂xf]−1.All the quantities in (6) are evaluated on the saddle tra-jectory satisfying the initial conditionsx˙2i = 2(E − Ey) , y˙2i + ω2y2i = 2Ey , (7)where x˙i ≡ x˙(ti). In deriving (6) we assumed that thesaddle configuration is unique; this is indeed the case forthe model (2).The amplitude (6) is to be inserted into the formulafor the tunneling probability,P = lim∆t→∞1∆t∫xf>0dxf |A|2 =∫dyf x˙f |A|2 , (8)where in the second equality we cancelled the diver-gence of the integral originating from the infinite regionxf → +∞ by writing 1∆t∫dxf = x˙f . Note that this op-eration is legitimate only if the boundary conditionxf → +∞ as tf → +∞ . (9)is satisfied. One substitutes (6) into (8) and evaluatesthe saddle-point integral over yf . One arrives at the ex-pression (1) withF = 2 Im(Scl +B) , A = ~1/2ωx˙f√2π|D1|2D2, (10)where D2 = 2 Im∂2Scl∂y2f. The saddle–point equation de-termines the remaining boundary condition for the tun-neling trajectory,Im y˙f = Im yf = 0 . (11)Note that reality of the total energy together with (9),(11) imply that x˙f is also real.One solves numerically the classical equations of mo-tion with the boundary conditions (7), (9), (11) and dis-covers [7] that tunneling solutions [14] satisfying all theconditions exist only at E < Ec(Ey). Still, one can findsolutions at Ec < E < Eb by imposing, instead of (9),the reality condition Im x˙f = Imxf = 0. But the re-sulting trajectories never come out from the interactionregion: they end up oscillating on top of the potentialbarrier. These solutions at t→ +∞ become precisely thesphalerons we referred to above. The formula (10) for theprefactor is not applicable in this case: the quantities D1,D2 entering into it describe linear response of the tun-neling solutions to small perturbations of the boundaryconditions and are ill-defined due to the instability of thesolutions.3To deal with this situation, we make the followingsteps. First, we introduce a functional Tint[x(t)] definedon classical paths. The choice for Tint[x(t)] is restrictedby three requirements: (a) it must be real and positive–definite on real paths, (b) it must be finite on paths end-ing up in the out–region, (c) it must diverge on pathswhich stay forever in the interaction region. Overall,the functional Tint[x(t)] should roughly measure the timespent by the particle in the interaction region. For themodel (2) the simplest choice is Tint[x(t)] =∫dt f(x(t)),where the function f(x) > 0 vanishes at x→ ±∞.Second, we restrict the path integral (5) to paths stay-ing fixed time in the interaction region, Tint[x(t)] = τ .This eliminates unstable trajectories from the domain ofintegration. The full propagator is then recovered by in-tegrating over τ . This program is realized by insertingthe unity1=∫dτ δ(Tint[x(t)]− τ)=∫dτ∫−i∞+i∞idǫ2π~eǫ(τ−Tint[x])/~into (5) and changing the order of integration. We obtain,〈xf |e−iH∆t/~|xi〉=∫dτ∫−i∞+i∞idǫ2π~eǫτ/~∫x(tf )=xfx(ti)=xi[dx] ei(S[x]+iǫTint[x])/~ .(12)One observes that the integral over [dx] in (12) is thesame as in (5) up to the substitutionS 7→ Sǫ = S + iǫTint . (13)Therefore, one can follow the steps leading from (5) to(6) with the resultA =∫dτ∫−i∞+i∞idǫ2π~eǫτ/~√ω2πD1,ǫei(Sclǫ +Bǫ)/~+iπ/4 .Importantly, the semiclassical trajectory xǫ(t) here is asolution to the equations of motion obtained from themodified action Sǫ. By construction, it spends a finitetime interval in the interaction region, and thus is stable.The integral over ǫ is saturated by the saddle–point ǫ(τ)which is implicitly defined by the condition Tint[xǫ] = τ .The latter follows from the relation d(Sclǫ + Bǫ)/dǫ =iTint[xǫ]. Note that the saddle-point value ǫ(τ) is notnecessarily purely imaginary; one should be careful topick the one with Re ǫ(τ) ≥ 0 in order to ensure theconvergence of the path integral in (12).One proceeds by substituting the expressions for theamplitude and its complex conjugate into the tunnelingprobability and performing integration over the final co-ordinates. This leaves the integral over two interactiontimes, τ and τ ′, coming from the amplitude and the com-plex conjugate amplitude. It is convenient to change theintegration variables to τ+ = (τ + τ′)/2, τ− = τ − τ ′.The integral over τ− is, again, saturated by the saddlepoint; one uses the formula d(Sclǫ + Bǫ − iǫτ)/dτ = −iǫin deriving the saddle-point condition. One obtainsP =∫dτ+ωx˙ǫ,f√−dǫ/dτ+π√2|D1,ǫ|2D2,ǫe−2(Im(Sclǫ +Bǫ)−ǫτ+)/~, (14)where the integral is performed over the real axis, and thefunction ǫ(τ+) is defined by the folowing implicit relationTint[xǫ] + Tint[x−ǫ] = 2τ+ . (15)Note that the solution to (15) is generically real for realτ+: the equations of motion following from Sǫ lead tox−ǫ∗ = x∗ǫ , which for real ǫ implies that the l.h.s of (15)is, indeed, real.So far, we did not refer to the particularities of thesphaleron–driven tunneling. The step where they be-come important is integration over τ+. For both directand sphaleron-driven tunneling, the integral (14) is dom-inated by the point ǫ(τ+) = 0 which corresponds to theoriginal unregularized trajectory. In the standard caseof direct tunneling the unregularized trajectory spendsa finite time interval τ+ in the interaction region; oneobtains the expressions (10) for the suppression expo-nent and prefactor by the saddle–point integration overτ+. In the sphaleron–driven case the interaction timecorresponding to ǫ = 0 is infinite. Thus, the integralis saturated by the end–point of the integration inter-val, τ+ → +∞. The tunneling probability in this case isdetermined by the behavior of the integrand in (14) atτ+ → +∞, that is, ǫ→ +0. Performing the integration,one obtains the formula (1) withF = limǫ→+0Fǫ , A =~1/2 limǫ→+0Aǫǫ√−4π dReTint[xǫ]dǫ, (16)where Fǫ and Aǫ are given by (10) evaluated on the ǫ-regularized solution xǫ. [A subtle point is that Fǫ =2(Im(Sclǫ + Bǫ) − ǫτ+) = 2 Im(S[xǫ] + Bǫ) is to be com-puted using the original action evaluated on the regu-larized solution.] Note an additional factor ~1/2 in theprefactor compared to the case of direct tunneling.Let us summarize our results. We have derived thefollowing prescription for the calculation of the tunnel-ing probability in the sphaleron–driven case. First, onereplaces the action of the system with the modified actionSǫ, (13), where ǫ > 0. Second, one finds the tunnelingsolution xǫ of the modified equations of motion. This so-lution interpolates between the in– and out–regions andis stable. One evaluates its suppression exponent Fǫ andprefactor Aǫ using the ordinary “direct tunneling” for-mulae. Third, one determines the suppression exponentand prefactor by taking the limit ǫ → +0 according to(16). Note that our prescription does not make use ofany particular properties of the illustrative model (2);4it is applicable in a large class of models exhibiting thephenomenon of sphaleron–driven tunneling.We checked the method of ǫ–regularization by applyingit to the model (2) and comparing the semiclassical re-sults with the exact suppression exponent and prefactorextracted from the numerical solution to the Schro¨dingerequation. The latter quantities were obtained by fittingthe dependence of the exact tunneling probability on ~with the formula~ logP(~) ≈ −F + γ2~ log ~ + ~ log A˜ , (17)where A˜ is independent of ~. For completeness, we alsoperformed the comparison in the regime of direct tunnel-ing. We set γ = 1 for the direct tunneling and γ = 2in the sphaleron–driven case. Figure 1 shows the de-pendences [15] F (E) and A˜(E) at fixed Ey = 0.05. Thesemiclassical and exact quantum–mechanical calculationsare in good agreement.Note that the quality of the fit (17) becomes worse asone approaches the transition point Ec between the twotunneling regimes. This is a manifestation of the break-down of the semiclassical approximation in the vicinityof this point. It is caused by the change in the depen- 0 1 2 3 4 5 6 71.2Ec10.80.60.40.20.050.20.10FEA ~(a)Sphaleron--driven tunnelingsuppression exponentprefactor 0 0.2 0.4Eb1.351.31.251.21.15Ec3210EF A ~(b)Direct tunnelingClassical transitionssuppression exponentprefactorFIG. 1: Comparison between the semiclassical (lines) and ex-act (points) results for the suppression exponent and prefactorin the cases of (a) direct and (b) sphaleron–driven tunneling.The comparison is performed in the model (2) with ω = 0.5,Ey = 0.05. The error bars represent the uncertainty of the fit(17).dence of the prefactor on ~. Indeed, A˜(E) diverges asE → Ec + 0, see Fig. 1b, which contradicts the con-tinuity of the exact tunneling probability. However, thesize of the vicinity where the semiclassical approximationbreaks down vanishes in the limit ~ → 0.Acknowledgements. We thank to F.L. Bezrukov,D.S. Gorbunov and V.A. Rubakov for discussions. Thiswork was supported in part by the RFBR grant 05-02-17363, the Grants of the Russian Science Support Foun-dation (D.L. and S.S.), the Fellowship of the “Dynasty”Foundation (A.P.), and the EU 6th Framework MarieCurie Research and Training network ”UniverseNet”(MRTN-CT-2006-035863). The numerical calculationswere performed on the Computational cluster of the The-oretical division of INR RAS.[1] O. Bohigas, S. Tomsovic and D. Ullmo, Phys. Rep.223, 43 (1993); S. Takada, P.N. Walker and M. Wilkin-son, Phys. Rev. A 52, 3546 (1995); A. Shudo, K.S.Ikeda, Phys. Rev. Lett. 74, 682 (1995); T. Van Voorhis,E.J. Heller, Phys. Rev. A 66, 050501(R) (2002);A.D. Ribeiro, M.A.M. de Aguiar, M. Baranger, Phys.Rev. E 69, 066204 (2004); S.C. Creagh, Nonlinearity 18,2089 (2005); Phys. Rev. Lett. 98, 153901 (2007).[2] C. Dembowski et al, Phys. Rev. Lett. 84, 867 (2000).[3] W.K. Hensinger et al, Nature (London) 412, 52 (2001).[4] D.A. Steck, W.H. Oskay and M.G. Raizen, Science 293,274 (2001).[5] W.H. Miller, Adv. Chem. Phys. 25, 69 (1974).[6] Tunneling in complex systems, edited by S. Tomsovic(World Scientific, Singapore, 1998).[7] F. Bezrukov and D. Levkov, arXiv:quant-ph/0301022; J.Exp. Theor. Phys. 98, 820 (2004) [Zh. Eksp. Teor. Fiz.125, 938 (2004)].[8] K. Takahashi and K.S. Ikeda, J. Phys. A 36, 7953 (2003);Europhys. Lett. 71, 193 (2005).[9] F. Bezrukov, D. Levkov, C. Rebbi, V.A. Rubakov andP. Tinyakov, Phys. Rev. D 68, 036005 (2003); D.G. Lev-kov and S.M. Sibiryakov, Phys. Rev. D 71, 025001(2005).[10] D.G. Levkov, A.G. Panin and S.M. Sibiryakov,arXiv:nlin.cd/0701063; arXiv:0704.0409 [quant-ph].[11] K. Takahashi and K.S. Ikeda, Phys. Rev. Lett. 97,240403 (2006).[12] Unstable semiclassical solutions in a different contextwere studied in P.B. Wilkinson et al., Nature (London)380, 608 (1996); S.C. Creagh and N.D. Whelan, Phys.Rev. Lett. 77, 4975 (1996); 82, 5237 (1998).[13] S. Levit and U. Smilansky, Ann. Phys. 103, 198 (1977).[14] These solutions are defined along a certain contour in thecomplex time plane which does not, in general, coincidewith the real time axis.[15] Due to computational limitations we were unable to ex-tract the prefactor A˜ from the solution of the Schro¨dingerequation at E < 0.56.

Unstable Semiclassical Trajectories in Tunneling

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