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Curve crossing in linear potential grids: the quasidegeneracy approximation

The quasidegeneracy approximation [V. A. Yurovsky, A. Ben-Reuven, P. S. Julienne, and Y. B. Band, J. Phys. B {\bf 32}, 1845 (1999)] is used here to evaluate transition amplitudes for the problem of curve crossing in linear potential grids involving two sets of parallel potentials. The approximation describes phenomena, such as counterintuitive transitions and saturation (incomplete population transfer), not predictable by the assumption of independent crossings. Also, a new kind of oscillations due to quantum interference (different from the well-known St\"uckelberg oscillations) is disclosed, and its nature discussed. The approximation can find applications in many fields of physics, where multistate curve crossing problems occur.


I. INTRODUCTION
The concept of curve crossing has many applications in the study of atomic collisions [1][2][3][4][5][6][7], excitations of atoms and molecules by nonstationary fields [8][9][10][11], Bose-Einstein condensates [12,13], and solid state physics [14,15]. Typical curve crossing problems are generally divided into two classes, R-dependent and t-dependent ones. The description of inelastic collisions, for example, involves crossings of coordinate-dependent potentials, and can therefore be treated as an R-dependent problem, described by a set of coupled second-order stationary Schrödinger equations. By the use of a commontrajectory approximation [1], R-dependent problems can be reduced to t-dependent ones. The latter class also naturally appears in the description of transitions due to non-stationary fields. Typical t-dependent problems involve the crossing of time-varying potentials, and their description requires a set of coupled first-order nonstationary Schrödinger equations.
Curve crossing problems are usually solved by using semiclassical approximations. Two-state crossing is described by the Landau-Zener (LZ) formula [16,17] (or one of its various modifications [18]), while multistate crossing is treated as a sequence of independent twostate crossings. Although semiclassical approaches are satisfactory for many applications (see, e.g. [6]), they fail to describe certain effects found recently in experiments, numerical calculations, and analytically-soluble potential models [5,[19][20][21][22].
The LZ formula forms an exact solution of the problem involving the crossing of two linear potentials infinitely diverging on both asymptotes. It gives good results even when local perturbations in the vicinity of the crossing are taken into account [9]. However, if the potentials retain a finite potential gap on an asymptote [20], have singularities [21,22], or are truncated [23], the transition probabilities deviate essentially from the LZ formula. Some interesting effects appear also in state crossing involving Bose-Einstein condensates, described by the nonlinear Gross-Pitaevskii equations (see [12]), instead of the linear Schrödinger equations.
The treatment of multistate curve crossing as a sequence of independent two-state crossings, commonly used in semiclassical approaches, fails to describe "counterintuitive" transitions (see [5,19]), in which the second crossing precedes the first one. In the case of a t-dependent linear grid (see [11,14,15,24,25]) consisting of two sets of mutually parallel potentials (see Fig. 1), counterintuitive transitions are exactly forbidden if the problem is defined on the infinite time interval (t ′ → ∞, t ′′ → −∞), as has been shown in Ref. [24]. However, the potentials in the grid infinitely diverge on both asymptotes, which is unphysical. We consider here a truncated linear grid, defined on a finite time interval [−t ′ , t ′′ ]. Such a truncation is relevant for application to transitions in time-dependent fields (e.g., single electronics [14,15]), since the field variation is actually finite.
The problem is studied here with the help of the "quasidegeneracy" approximation, introduced in [12] for the case when one of the sets consists of only one potential. This approximation treats a non-degenerate system with small potential gaps as a perturbed degenerate system (see [26]). A special form of the quasidegeneracy approximation was used also in Ref. [14]. This form is applicable to a linear grid consisting of two potentials in each set, with equal couplings between all states belonging to different sets, while only one of the sets is quasidegenerate. The method of Ref. [14] is actually a simplified form of the method of Ref. [19]. A linear grid in which one set of parallel potentials is exactly degenerate was considered also in Ref. [10], by using a method different from the quasidegeneracy approximation.
The quasidegeneracy approximation is generalized here to the case of a truncated linear grid with arbitrary number of potentials in both sets. In Sec. II we introduce a "decoupling" transformation, which approximately transforms the problem to a set of parallel twostate crossings. The transition amplitudes are calculated in Sec. III, and applicability criteria are presented in Sec. IV. Results are shown and discussed in Sec. V. A few preliminary results of this work have been presented in [27].

II. DECOUPLING TRANSFORMATION
Let us consider two sets of mutually parallel linear potentials. This problem can be easily reduced by a gauge transformation to the case of a set of horizontal potentials V j (j = 1, . . . , n 1 ) crossed by a set of slanted parallel linear potentials V n1+k + βt (k = 1, . . . , n 2 ) (see Fig. 1). The interactions between the states within each set of parallel potentials can be eliminated by a unitary transformation. Therefore, without loss of generality, we can describe the problem by the following system of coupled equations for the expansion coefficients ϕ j (t), (using a system of units in whichh = 1). The only non-vanishing coupling coefficients g jk involve pairs of crossed potentials. The problem is defined here on the The special case of n 2 = 1 has been considered in [19], by using a quasidegeneracy approximation. In order to generalize this approximation, let us perform a singular value decomposition (SVD) for the coupling matrix g jk , of the form This decomposition is well known in the theory of spline approximations (see, e. g., [28]). The two matrices with elements X lj and Y lk are unitary, and their rows are the eigenvectors of the quadratic matrices formed by products of the g jk and their hermitian conjugates: A transformation of the expansion coefficients ϕ j (t) using the matrices X lj and Y lk , leads to a new system of coupled equations Given a matrix g jk , its SVD is not unique, and may be chosen in such a way that its singular values g l are real and non-negative, and the non-diagonal potential elements V (a) ll ′ vanish when l > n and l ′ > n. When both parallel sets of potentials are degenerate, the matrices V ll ′ are diagonal, and the system (4) describes a set of n independent pairs of crossing potentials, n 1 −n separate horizontal potentials (not coupled to other channels), and n 2 − n separate slanted potentials. Since the transformation (3) partially eliminates the coupling between the states, hereafter it is called the "decoupling transformation". The channels described by coefficients a and b will be called the "decoupled channels".
In the non-degenerate case the non-diagonal elements of V (a) ll ′ lead to transitions between the decoupled channels. However, the magnitudes of these nondiagonal elements are bounded by the inequalities where the bandwidths of the potential sets are defined as Therefore, these transitions are negligible if the bandwidths of the two potential sets are small enough. (Appropriate applicability criteria are presented in Sec. IV below.) Neglecting the non-diagonal elements of V ll ′ , we obtain a zero-order-approximation system of equations for a l (t) and b l (t), which describes the same set of decoupled channels as the one that prevails in the case of degenerate potentials. Given an arbitrary matrix g jk , the transformation matrices X lj and Y lk cannot be generally expressed in an analytical form. Nevertheless, analytical expressions can be obtained in the specific case of a separable matrix g jk = ξ * j η k . In this case, one of the rows (the first, for definiteness) has the form and the other rows are orthogonal to the first one. The singular values can then be written as In this case, n = 1 and the transformed system consists of one pair of coupled potentials, together with n 1 − 1 horizontal, and n 2 − 1 slanted, separate potentials.
In the case of equal couplings g jk = g (independent of j and k ), , and g l = (n 1 n 2 ) 1/2 gδ l1 . The opposite situation (in which g l is independent of l) takes place in the case in which the coupling matrix g jk is proportional to a unitary matrix.

III. TRANSITION AMPLITUDES
The zero-order equations (8c) and (8d) representing separate channels have the simple analytical solutions The remaining equations (8a) and (8b) represent a set of n two-state linear curve-crossing problems. In the limit t ′ → ∞, t ′′ → ∞ the transition amplitude in each of these systems is given by the LZ formula. However, the solution of this problem converges to the asymptotic limit very slowly. We shall therefore use the exact solution of the linear two-state curve crossing problem, known since the pioneering work of Zener [17]. The two independent solutions A ml (t), B ml (t), with m = 1, 2, can be expressed in terms of the confluent hypergeometric function 1 F 1 (see [29]) as are, respectively, the LZ exponent for the two-state crossing and the position of the crossing point on the time scale.
The transition matrix S (l) , connecting the coefficients a can be expressed in terms of the fundamental solutions (12) in the form In our numerical calculations we expressed the confluent hypergeometric functions in terms of columbic wavefunctions, using the algorithm of [30] for their evaluation. When the original representation (1) is recovered by application of the transformation (3) one obtains the transition matrix S, defined by in the zero-order approximation, as where 1 ≤ j ≤ n 1 , 1 ≤ j ′ ≤ n 1 , 1 ≤ k ≤ n 2 , and 1 ≤ k ′ ≤ n 2 . This solution constitutes the quasidegeneracy approximation. Whenever S (l) bb are l-independent, and n = n 1 (or n = n 2 ), transitions between states within the corresponding set of n 1 (or n 2 ) parallel potentials become forbidden due to the unitarity of matrices X lj and Y jk . Such an effect may take place if the couplings g l are close in magnitude or very small. If n < n 1 (or n < n 2 ) such transitions vanish only at low couplings, in which case |S and analogous expressions for ∆S ll ′ , obtained by replacing a with b everywhere in Eq. (19).
An overestimate for these amplitudes can be obtained by substituting a where the bandwidths of the potential sets ∆V 1 and ∆V 2 are defined by Eq. (7). However, in certain situations less stringent criteria may exist, as can be shown by the use of approximate expressions for the unperturbed wavefunctions [solutions of Eqs. (8)]. Such approximate expressions can be obtained in two limiting cases. The first one is the asymptotic case, in which the bounds −t ′ and t ′′ lie far outside the two-state transition ranges g l /β, i. e., In this case, an asymptotic expansion of the confluent hypergeometric function (see [29]) on the the left-hand asymptote t ′ + t l > −t + t l ≫ g l /β yields and on the right-hand asymptote Whenever l > n, Eqs. (22) and (23) become exact [see Eqs. (11)]. Hereafter one should set λ l = 0 if l > n.
The first-order corrections to the amplitudes (19) can be therefore estimated as for the horizontal set, and a similar expression, with b replacing a, for the slanted set. Finally, using Eq. (6) one can write the applicability criteria in the form Let us consider now the second limiting case, in which both boundaries −t ′ and t ′′ lie way inside the two-state transition ranges g l /β, i. e., In addition, let λ l ≫ 1, in order to obtain an adiabatic evolution. Within the range defined by Eq. (26), the adiabatic energies are approximately V (a) ll ± g l , and a (0) Substitution of Eqs. (27) in Eq. (19), taking into account Eq. (6), gives the applicability criteria Criteria combining the cases (20), (25), and (28) can be written with the help of Eq. (13) as the single expression These criteria allow for an interpretation that stems from the viewpoint of the uncertainty principle. Equation (29) means that the potentials become indistinguishable within a limited time interval. The second term in the right-hand side of Eq. (29) describes a broadening of the allowed uncertainty as the coupling increases.

V. RESULTS AND DISCUSSION
In the limiting case of a linear grid defined on the infinite time interval −∞ < t < ∞, some transitions become forbidden (see [24]). An example of such transitions is shown in Fig. 1, in which two time-independent potentials are shown crossed by three parallel time-slanted potentials. The forbidden transitions, such as 2 → 1, 3 → 4, 3 → 5, and 4 → 5, are called counterintuitive, since in order to treat them as a sequence of independent twostate crossings, one has to assume a motion backwards in time.
Counterintuitive transitions can nonetheless occur, as has been proven in numerical calculations involving crossings of nonlinear potentials [5], and in uses of the quasidegeneracy approximation for truncated and piecewise linear problems, involving a set of horizontal potentials, crossed by one slanted potential [19]. Such transitions are present in truncated linear grids as well, since the transformations (3) connect the initial and final states to all the decoupled channels.
Hereafter we shall demonstrate the application of the quasidegeneracy approximation to a particular example. Consider the model of a linear grid with n 1 = n 2 = 2, V 1 = V 3 = −∆V /2, and V 2 = V 4 = ∆V /2 (recalling that V 3 and V 4 are the time-independent parts of the slanted potentials). Let the coupling matrix have one of the two special forms, either with integer values of m, or the equal-coupling form, with All the following calculations are performed for the slope β = 1. The results can be readily expanded to other β values by the substitutions g/ √ β → g, ∆V / √ β → ∆V , and t √ β → t. Figure 2 presents the dependence of counterintuitive transition probabilities on the coupling strength g 0 for two cases: an exactly degenerate one (∆V = 0), and a one in which ∆V (t ′ + t ′′ ) = 0.5, on the verge of the validity criteria (29). At low values of g 0 the amplitudes S  Fig. 2, where g 1 = 2g 0 and g 2 = 0.38g 0 ). In the case of a separable matrix (see the plot for m = 0 in Fig. 2, where g 1 = 2.03g 0 and g 2 = 0), such transitions persist even in the limit of high coupling strength. It is worth noting that even a change of the phase of one element of the coupling matrix transforms a separable matrix to a non-separable one, and therefore changes the behavior of the transition probability at high values of the coupling strengths.
Counterintuitive transitions persist at finite values of the potential gap ∆V as well (see Figs. 2b and 3). As one can see, the higher is the coupling strength, the better are the results of the quasidegeneracy approximation [in agreement with the criteria Eq. (29)]. At low coupling strengths the predictions of the quasidegeneracy approximation are correct as long as (t ′ + t ′′ ) ∆V ≤ 0.2 (see 3a), while at high coupling strengths they are correct as long as (t ′ + t ′′ ) ∆V ≤ 0.2λ (see 3b).
Probabilities of counterintuitive transitions (see Fig.  3b) and other transitions (see Fig. 4) demonstrate an oscillating pattern in their dependence on the potential gap. The nature of these oscillations is different from the well-known Stückelberg oscillations (see Ref. [4]), which may be present only in transitions including two or more interfering "intuitive" paths. (Such paths exist in the transitions from 1 or 4 to 2 or 3 in the case presented in Figs. 3 and 4.) The period of the Stückelberg oscillations is ∆V /β; i.e., it is dependent on the potential gap ∆V but independent of the time interval t ′ +t ′′ . These properties, as well as the magnitude of the Stückelberg oscillation period, are not in agreement with the behavior of the oscillations presented in Figs. 3b and 4.
The quasidegeneracy approximation relates the oscillations reported here to the interference of the terms in Eqs. (18), corresponding to different decoupled channels. The dependence on ∆V is due to exponents in Eqs. (12) and in the second sum of each of the two equations (18a) and (18b). The oscillation period in ∆V is 2πρ/ (t ′ + t ′′ ), is the ratio of the potential gaps in the original and decoupled representations. For the coupling matrix (30) we have ρ = 5.6, 5.3, 4.0, and 2.5 for m = 0, 1, 2, and 3, respectively, which explains the variation of the oscillation period with m in Fig. 3. It is worth noting that in Fig There is still another kind of oscillations possible. It has been demonstrated in the case of a truncated twostate linear curve crossing, in which oscillations may show up as a function of each of the two truncation times (see [23]). In principle, such oscillations should also appear in our model in the ∆V dependence, since the crossing points move as the potential gap is varied. However, the period of these oscillations, too, is too large to show up in Figs. 3 and 4.
In the limit of high coupling strengths or slow potential variation (λ ≫ 1) the semiclassical approach of independent crossings predicts non-vanishing transitions only for one final state per a given initial state. If n 1 = n 2 all the non-vanishing transitions lead from one set of the parallel potentials to another set, leading to a complete population transfer between the sets. This property was used in a recent proposal of single-electronics devices, based on transitions between quantum dots (Refs. [14,15]). In contrast, the quasidegeneracy approximation predicts more non-vanishing transitions (see Figs. 3 and 4). In the case of n < n 1 = n 2 , a finite probability may remain for transitions within the same set of the parallel potentials (see Figs. 3, 4b, and 4c), leading to an incomplete population transfer. This effect is similar to the effect of incomplete optical shielding in ultracold atom collisions [5,6]. The effect of incomplete population transfer may interfere with the operation of the single-electronics devices mentioned above.

VI. CONCLUSIONS
Equations (18) describe the transition amplitudes in a truncated linear potential grid, whenever the applicability criteria (29) are observed. The results can be applied also to a more general case, in which the grid may be broken into well-separated groups of quasidegenerate crossings. In this case the transition amplitudes can be represented as products of the transition amplitudes given by Eqs. (18) for the quasidegenerate groups. Thus the approximation can be used in a wide variety of physical problems in which a multistate curve crossing occurs.