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RAPID COMMUNICATIONS PHYSICAL REVIEW A 76, 010102共R兲 共2007兲 Quantum interference in the classically forbidden region: A parametric oscillator 1 M. Marthaler1,2 and M. I. Dykman1 Department of Physics and Astronomy, Michigan State University, East Lansing, Michigan 48824, USA 2 Institut für Theoretische Festkörperphysik and DFG-Center for Functional Nanostructures (CFN), Universität Karlsruhe, D-76128 Karlsruhe, Germany 共Received 20 March 2007; published 20 July 2007兲 We study tunneling between period-2 states of a parametrically modulated oscillator. The tunneling matrix element is shown to oscillate with the varying frequency of the modulating field. The effect is due to spatial oscillations of the wave function and the related interference in the classically forbidden region. The oscillations emerge already in the ground state of the oscillator Hamiltonian in the rotating frame. DOI: 10.1103/PhysRevA.76.010102 PACS number共s兲: 03.65.Xp, 74.50.⫹r, 05.45.⫺a, 05.60.Gg Nonlinear micro- and mesoscopic vibrational systems have attracted much interest in recent years. In such systems damping is often weak, and even a comparatively small resonant field can lead to bistability, i.e., to coexistence of forced vibrations with different phases and/or amplitudes. Quantum and classical fluctuations cause transitions between coexisting vibrational states. The transitions are not described by the conventional theory of metastable decay, because the states are periodic in time and the systems lack detailed balance. Experimentally, classical transition rates have been studied for such diverse vibrational systems as modulated trapped electrons 关1兴, Josephson junctions 关2兴, nano- and micromechanical oscillators 关3–5兴, and trapped atoms 关6兴, and the results are in agreement with theory 关7,8兴. Currently much experimental effort is being put into reaching the quantum regime 关9,10兴. In this regime tunneling between coexisting classically stable periodic states should become important, for weak dissipation. It was first studied for a resonantly driven oscillator, where a semiclassical analysis 关11兴 made it possible to find the tunneling exponent in a broad parameter range 关12兴. Tunneling is particularly interesting for a parametrically modulated oscillator. Here, the coexisting classical periodic states have period 2F, where F is the modulation period. Such period-2 states are identical except that the vibrations are shifted in phase by . Therefore the corresponding quantum states 共Floquet states兲 are degenerate. Tunneling should lift this degeneracy, as for a particle in a symmetric static double-well potential. Earlier the tunneling matrix element was found 关13兴 for modulation at exactly twice the oscillator eigenfrequency 0. Recently the tunneling exponent was obtained in a general case where the modulation frequency F = 2 / F is close to 20 关14兴. In this paper we show that tunneling between period-2 states of a parametrically modulated oscillator displays unexpected features. We find that the tunneling matrix element oscillates with varying F − 20, periodically passing through zero. These oscillations are accompanied by and are due to spatial oscillations of the wave function in the classically forbidden region. For resonant modulation, 兩F − 20兩 Ⰶ F, and for a small amplitude of the modulating field F the oscillator dynamics is well described by the rotating wave approximation 共RWA兲 关15兴. The scaled RWA Hamiltonian ĝ as a function of the 1050-2947/2007/76共1兲/010102共4兲 oscillator coordinate Q and momentum P in the rotating frame is independent of time. In a broad parameter range it has a symmetric double-well form shown in Fig. 1. The minima correspond to the classical period-2 states, in the presence of weak dissipation. Respectively, of utmost interest are tunneling transitions between the lowest single-well quantum states of ĝ. A simple model of a nonlinear oscillator that describes many experimental systems, cf. Refs. 关1–6兴, is a Duffing oscillator. The Hamiltonian of a parametrically modulated Duffing oscillator has the form 1 1 1 H0 = p2 + 共20 + F cos Ft兲q2 + ␥q4 . 2 2 4 共1兲 For F close to 20 and for comparatively small F, 1 2 ␦ = F − 0, 兩 ␦ 兩 Ⰶ 0, F Ⰶ 20 , 共2兲 even where the oscillator becomes bistable its nonlinearity remains relatively small, 兩␥具q2典兩 Ⰶ 20. For concreteness we set ␥ ⬎ 0; the results for ␥ ⬍ 0 can be obtained by replacing ␦ → −␦ in the final expressions. 0.6 g 0.0 0.5 -0.6 -2.0 0.0 -1.0 P 0.0 Q 1.0 -0.5 2.0 FIG. 1. 共Color online兲 The scaled effective Hamiltonian of the oscillator in the rotating frame g共Q , P兲, Eq. 共5兲, for = 0.5. The minima of g共Q , P兲 correspond to the period-2 vibrations. The eigenvalues of ĝ give scaled oscillator quasienergies. 010102-1 ©2007 The American Physical Society RAPID COMMUNICATIONS PHYSICAL REVIEW A 76, 010102共R兲 共2007兲 M. MARTHALER AND M. I. DYKMAN To describe a weakly nonlinear oscillator it is convenient to make a canonical transformation from q and p to the slowly varying coordinate Q and momentum P, U†qU = Cpar关P cos共Ft/2兲 − Q sin共Ft/2兲兴, U† pU = − CparF关P sin共Ft/2兲 + Q cos共Ft/2兲兴/2, 共3兲 where Cpar = 共2F / 3␥兲1/2 and 关P,Q兴 = − i, = 3␥ប/FF . 共4兲 The dimensionless parameter plays the role of ប in the quantum dynamics in the rotating frame 关14兴. The transformed oscillator Hamiltonian has the form 共F2 / 6␥兲ĝ, where ĝ ⬅ g共Q , P兲, 1 1 1 g共Q, P兲 = 共P2 + Q2兲2 + 共1 − 兲P2 − 共1 + 兲Q2 4 2 2 ±共Q兲 = 1 冑2 关l共Q兲 ± l共− Q兲兴, 共7兲 where l共Q兲 is the “single-well” wave function of the left well of g共Q , P兲 in Fig. 1. It is maximal at the bottom of the well Ql0 = −共1 + 兲1/2 and decays away from the well. To the leading order in , the corresponding lowest eigenvalue of ĝ is gmin + gq, where gmin = −共1 + 兲2 / 4 is the minimum of g共Q , P兲 and gq = 共 + 1兲1/2 is the zero-point energy. The wave function l共Q兲 is particularly simple for ⬍ 0. In the classically forbidden region between the wells, 兩Q兩 ⬍ 兩Ql0兩, it has the form l = C关− i Pg兴−1/2 exp关iS0共Q兲/兴, 共8兲 where S0共Q兲 is given by the equation g共Q , QS0兲 = gmin + gq, S0共Q兲 = 共5兲 冕 Q P−共Q⬘兲dQ⬘ , Ql0+Lq ˜ 兲1/2兴1/2 , P±共Q兲 = i关1 + Q2 − ± 2共Q2 − 关we use here a more conventional notation g共Q , P兲 instead of g共P , Q兲 used in Ref. 关14兴兴. The terms ⬀exp共±inFt兲 with n 艌 1 in ĝ have been disregarded. The time-independent operator ĝ is the scaled oscillator Hamiltonian in the rotating frame. Its eigenvalues multiplied by F2 / 6␥ give oscillator quasienergies, or Floquet eigenvalues. Formally, ĝ is a Hamiltonian of an auxiliary stationary system with variables Q , P, and the eigenvalues of ĝ give the energies of this system. The operator ĝ depends on one parameter We keep in S0 only the contribution from the branch P−共Q兲, because P−共Q兲 is zero on the boundary of the classically forbidden range Ql0 + Lq. For − Ⰷ and 兩Ql0 + Lq兩 ⬎ 兩Q兩 the action S0共Q兲 is purely imaginary. The wave function l共Q兲 monotonically decays with increasing Q. The prefactor in the wave function 共8兲 is determined by the complex classical speed of the oscillator = 2F␦/F. ˜ 兲1/2 . Pg = 2P−共Q兲共Q2 − ˜ = − g q, 共6兲 For ⬎ −1, g共Q , P兲 has two minima located at P = 0, Q = ± 共 + 1兲1/2. For 艋 1 the minima are separated by a saddle at P = Q = 0, as shown in Fig. 1. When friction is taken into account, the minima become stable states of period-2 vibrations. The function g共Q , P兲 is symmetric as a consequence of the time translation symmetry: the change 共P , Q兲 → 共−P , −Q兲 corresponds to shifting time in Eq. 共3兲 by the modulation period F. We assume the effective Planck constant to be the small parameter of the theory, Ⰶ 1. Then the low-lying eigenvalues of ĝ form doublets. Splitting of the doublets is due to tunneling between the wells of g共Q , P兲. Since g共Q , P兲 = g共−Q , −P兲 is symmetric, the problem of level splitting seems to be similar to the standard problem of level splitting in a double-well potential 关16兴. As in this latter case, we will analyze it in the WKB approximation. The major distinction of the present problem comes from the difference between the structure of g共Q , P兲 and the Hamiltonian considered in Ref. 关16兴. The momentum P共Q ; g兲 as given by equation g共Q , P兲 = g has four branches, with both real and imaginary parts in the classically forbidden region of Q. This leads to new features of tunneling and requires a modification of the method 关16兴. We will consider splitting ␦g of the two lowest eigenvalues of ĝ. Because of the symmetry, the corresponding wave functions ±共Q兲 are 1/2 −1/4 Lq = /g1/2 . q ⬅ 共 + 1兲 共9兲 共10兲 The normalization constant C in Eq. 共8兲, C = 关共 + 1兲/兴1/4 exp共− 1/4兲, 共11兲 is obtained by matching, in the range Lq Ⰶ Q − Ql0 Ⰶ 兩Ql0兩, Eq. 共8兲 to the tail of the Gaussian peak of l共Q兲, which is centered at Ql0. We are most interested in the parameter range Ⰷ where tunneling displays unusual behavior. For such the ˜. momentum P−共Q兲 becomes complex in the range 兩Q兩 ⬍ This means that the decay of the wave function is accompanied by oscillations. To correctly describe them we had to keep corrections ⬀gq in Eq. 共9兲. We first rewrite Eq. 共9兲 in the form 冋 ˜ 兲1/2 − P−共Q兲 ⬇ i 1 − 共Q2 − gq/2 ˜ 兲1/2 1 − 共Q2 − 册 . 共12兲 Equation 共12兲 applies for Q − Ql0 Ⰷ Lq. It is seen that P−共Q兲 has two branching points inside the classically forbidden re˜ 1/2. The WKB gion. The closest to Ql0 is the point Qbr = − ˜ 1/2. The wave approximation breaks down for small Q + function in this region can be shown to be proportional to ˜ 1/2兲共2 ˜ 1/2 / 2兲1/3). Therefore l osAiry function Ai(−共Q + ˜ 1/2. cillates with Q for positive Q + In contrast to the standard WKB theory of the turning point, the prefactor in l contains two factors that experience ˜ 1/2, see Eqs. 共8兲 and 共10兲. The full solution in branching at − 010102-2 RAPID COMMUNICATIONS PHYSICAL REVIEW A 76, 010102共R兲 共2007兲 QUANTUM INTERFERENCE IN THE CLASSICALLY… FIG. 2. 共Color online兲 The wave function of the ground state in the left well l共Q兲 in the oscillation region for = 0.09 and = 0.5. The solid line shows explicit expressions 共13兲–共15兲, the dashed line shows numerical results. Inset: l共Q兲 near its second zero with higher resolution. ˜ 1/2 the oscillation region can be obtained by going around − in the complex plane following the prescription 关16兴. For ˜ 1/2 this gives 2/3 Ⰶ Q + FIG. 3. 共Color online兲 Scaled matrix element of tunneling between period-2 states as a function of the scaled detuning of the modulation frequency from twice the oscillator eigenfrequency. The solid lines show explicit expression 共16兲, the dashed lines show the result of numerical calculations. Inset: a higher-resolution plot of 兩␦g兩 / 2 vs near the zero of ␦g at = 6. The data refer to = 0.09. ␦g = − 2兵2共1 − 兲l共0兲l⬘共0兲 − 2关l共0兲l共0兲 + l⬘共0兲l⬙共0兲兴其 l ⬇ 2C兩 Pg兩−1/2 exp关− Im S0共Q兲/兴cos ⌽共Q兲, or, with account taken of Eq. 共13兲, ⌽共Q兲 = ⌽1共Q兲 + ⌽2共Q兲. 共13兲 ␦g = The term Im S0共Q兲 in the amplitude of the wave function 共13兲 is determined by Eq. 共9兲. The phase ⌽共Q兲 has two terms. The term ⌽1共Q兲 comes from the exponential factor in the WKB wave function 共8兲, ⌽1共Q兲 = −1 冕 2⌽1共0兲 = 共−1 − 1兲/2 Re P−共Q兲dQ, 共14兲 where Re P−共Q兲 is given by Eq. 共12兲 in which we set 共Q2 ˜ 兲1/2 → i共 ˜ − Q2兲1/2; therefore Re P−共Q兲 ⬎ 0. It is simple to − write ⌽1 and Im S0共Q兲 in explicit form. The term ⌽2共Q兲 in Eq. 共13兲 comes from the prefactor in l共Q兲, Eq. 共8兲, ⌽2共Q兲 ⬇ A = 共 + 1兲1/2 + ln兵−1/2关1 + 共 + 1兲1/2兴其, Q ˜ 1/2 − 冉 161/2共 + 1兲5/4 −A/ e cos关2⌽1共0兲兴, 共兲1/2 1 − Q2 arcsin 2 1 + − Q2 冊 1/2 − . 4 共15兲 Decay and oscillations of the wave function described by Eq. 共13兲 are compared in Fig. 2 with the results of a numerical solution of the Schrödinger equation ĝ = g. The leftwell wave function was obtained numerically as a sum of the two lowest-eigenvalues solutions, cf. Eq. 共7兲. In this calculation the basis of 120 oscillator Fock states was used. A good agreement between analytical and numerical results is seen already for not too small = 0.09. The above solution allows us to find the tunnel splitting ␦g = g− − g+ of the symmetric and antisymmetric states 共7兲. Following the standard approach for a symmetric doublewell potential 关16兴 we multiply the Schrödinger equations for the involved states ĝl = gll and ĝ± = g±± by *± and *l , respectively, integrate over Q from −⬁ to 0, and subtract the results. This gives 共 Ⰷ 兲. 共16兲 Clearly, ␦g may be positive or negative, that is, the symmetric state may have a lower or higher quasienergy than the antisymmetric state. The dimensional splitting 共F2 / 6␥兲兩␦g兩 gives twice the matrix element of tunneling between period-2 states of the oscillator. This matrix element has an exponential factor exp共−A / 兲 关14兴. In addition, it contains a factor oscillating as a function of the scaled frequency detuning / = 62F共F − 20兲 / 3␥ប. The oscillation period is ⌬共 / 兲 = 4. These oscillations are shown in Fig. 3. The oscillations of ␦g result from the wave function oscillations in the classically forbidden region. This can be seen from the analysis of l共Q兲 near the positive-Q boundary ˜ 1/2. The wave function for Q of the oscillation region, Q = ˜ 1/2 Ⰷ is a combination of the WKB waves with imagi− ˜ 兲1/2兴. The coefficients in nary momenta P±共Q兲 ⬇ i关1 ± 共Q2 − this combination can be found in a standard way 关16兴. They ˜ 1/2兲. Only the wave with are determined by the phase ⌽共 P−共Q兲 contributes to the tunneling amplitude, since P+ re˜ 1/2兲 mains imaginary in the right well of g共Q , P兲. For ⌽共 = 共4n − 3兲 / 4 this wave has zero amplitude, leading to ␦g ˜ 1/2兲 = 2⌽1共0兲 − / 4, we immediately = 0. By noting that ⌽共 obtain from Eq. 共16兲 that ␦g = 0 for = 2n with integer n, in agreement with Fig. 3. The occurrence of spatial oscillations of the ground state wave function of the scaled Hamiltonian ĝ does not contra- 010102-3 RAPID COMMUNICATIONS PHYSICAL REVIEW A 76, 010102共R兲 共2007兲 M. MARTHALER AND M. I. DYKMAN dict the oscillation theorem, because ĝ is not a sum of the kinetic and potential energies and is quartic in P. The motion described by the Hamiltonian g共Q , P兲 is classically integrable. Respectively, the quantum problem is different from dynamical tunneling in classically chaotic systems 关17–19兴; the effect we discuss has not been considered for such systems, to the best of our knowledge. The effect is also qualitatively different from photonassisted or suppressed tunneling in systems with stationary double-well potentials: our oscillator has a single-well potential, the bistability is a consequence of resonant modulation, and the Hamiltonian ĝ is independent of time. On the other hand, there is a remote similarity between the oscillations of the tunneling matrix element for period-2 states and for electron states in a double-well potential in a quantizing magnetic field 关20兴. However, not only is the physics different, but our approach is also different from that in Ref. 关20兴; in particular, it makes it possible to find ␦g analytically. The approach can be extended also to a resonantly driven Duffing oscillator, where the RWA Hamiltonian has a structure similar to Eq. 共5兲 关7,12兴. Tunnel splitting can be observed by preparing the system in one of the period-2 states and by studying interstate oscillations, cf. Refs. 关18,19兴. This requires that the tunneling rate 共␦ / 2兲兩␦g兩 exceed F / 4Q, where Q is the oscillator quality factor. The splitting sharply increases with increasing . It will be shown separately that for comparatively large 共but still for 兩␦g兩 Ⰶ gq兲 the RWA applies and relaxation remains small provided ␦g2 Ⰷ C / Q with C ⱗ 1. Our RWA numerical results indicate that ␦g still oscillates with for = 0.25− 0.3 and is well described by Eq. 共16兲 for ⲏ 2. The local peak of 兩␦g兩 for = 0.3 and the characteristic extremum of d2␦g / d2 for = 0.25 occur where 兩␦g兩 ⬇ 0.01. 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Lifshitz, Quantum Mechanics. Nonrelativistic Theory, 3rd ed. 共Butterworth-Heinemann, Oxford, 1981兲. 关17兴 S. Tomsovic and D. Ullmo, Phys. Rev. E 50, 145 共1994兲. 关18兴 W. K. Hensinger et al., Nature 共London兲 412, 52 共2001兲. 关19兴 D. A. Steck, W. H. Oskay, and M. G. Raizen, Science 293, 274 共2001兲. 关20兴 J. K. Jain and S. Kivelson, Phys. Rev. B 37, 4111 共1988兲. We are grateful to V. N. Smelyanskiy and F. Wilhelm for stimulating interactions. This research was supported in part by the NSF through Grant No. PHY-0555346. 010102-4