PHYSICAL REVIEW A VOLUME 50, NUMBER 1 JULY 1994 Role of phase coherence in the transition dynamics of a periodically driven two-level system Yosuke Kayanuma Department of Physics, Faculty of Science, Tohoku University, Sendai 980, Japan (Received 17 February 1994) Some aspects of quantum tunneling of a particle in a double-we11 potential periodically driven by an external force are studied within the two-level approximation. A closed expression for the temporal evo- lution of the occupation probability is obtained in the limit of large-amplitude oscillation by the transfer-matrix formalism. The mechanism of coherent destruction of tunneling found by Grossmann et al. [Phys. Rev. Lett. 67, 516 (1991)] is made clear from a viewpoint of interference at periodic level crossings. PACS number(s): 03. 65. — w, 33. 80. Be, 73. 40. Gk H(t) =Ho+Sx cos(tot ), 2 Ho= + V(x), 2M (2) where V(x) is a symmetric double-well potential, and S is a coupling constant with an oscillating field of frequency N. In this connection, Grossmann and et al. [5, 6] found a peculiar behavior in the tunneling dynamics of a driven system which they termed "a coherent destruction of tun- neling:" A Gaussian wave packet initially located in one of the potential wells never transfers to the other well, as if the quantum tunneling is frozen by the periodic modu- lation. They reported that this phenomenon occurs only for parameter values of S and ~ restricted to one- dimensional manifolds in two-dimensional parameter plane. Earlier than this, Lin and Ballentine [3] had no- ticed that the tunneling probability is highly enhanced due to the periodic modulation on the basis of numerical calculation for just the same models as that of Grossmann et al. but in a different parameter region. These observations strongly suggest that the tunneling dynamics in the driven system is governed by a mecha- nism quite different from that of the static potential sys- tem. The 1ow-lying eigenstates of the unperturbed Hamil- tonian Ho form nearly degenerate doublets which are composed of the linear combinations of pairs of low-lying states localized in the left and the right wells, respective- ly, and split by the barrier tunneling. Under the condi- tion that the tunnel splitting is small enough, and the The quantum dynamics of a particle in a double-well potential periodically driven by an external force has been a subject of considerable interest in recent years. The attenuation has mainly focused on the possible chaotic behavior or "quantum chaos" that the system ex- hibits, and its relationship with that of its classical coun- terpart [1-3]. The effect of the external modulation on quantum tunneling through the classically impenetrable region has been another, related, theme of investigation [4 — 6). An archetypal model Hamiltonian is given in the form modulation amplitude of the energies of the left and right wells is much smaller than the representative excitation energy in a single well, the transition dynamics can well be described by the two-level model for the lowest dou- blet. The suppression of tunneling has been discussed within the two-level model by applying Floquet formal- ism [6, 7]. The purpose of the present report is to propose a slightly different point of view for this problem, from which one can obtain further insight into the mechanism of the suppression of tunneling. Let us consider a two-level Hamiltonian, HTL(t) =-, ' A cos(tot)(li & & ll — 12& &2I) +&(ll & &2I+ I2& & ll), (3) The unit fi= 1 is used here and hereafter. The probability that the system is in I2& at time t under the condition that it starts from I 1 &at t =0 will be denoted as P(t). Al- though it is an easy matter to calculate P(t) numerically by direct integration or by the technique of Floquet map- ping, our purpose is to obtain analytical expressions. This can be done in some limiting cases in the parameter space (b, A, to). 1. Limit of rapid oscillation When the condition 6 &&cu where I 1 & and I2& represent, say, the left and the right localized states, respectively. For the state vector written in the form IV(t)&=C, (t)exp[ i( A— /2 co) sin( tot)] ll & + Cz(t)exp[i( A /2co)sin(tot )] I2 &, the Schrodinger equation is given by i C, (t) = b, exp[i( A /to)sin(tot )]C2(t), . d t (4) i C2(t) = — 6 exp[ i ( A /to— )sin(tot ) ]C, (t) . . d 1050-2947/94/50(1)/843(3)/$06. 00 50 843 1994 The American Physical Society