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Analytic calculation of nonadiabatic transition probabilities from monodromy of differential equations

The nonadiabatic transition probabilities in the two-level systems are calculated analytically by using the monodromy matrix determining the global feature of the underlying differential equation. We study the time-dependent 2x2 Hamiltonian with the tanh-type plus sech-type energy difference and with constant off-diagonal elements as an example to show the efficiency of the monodromy approach. The application of this method to multi-level systems is also discussed.


Introduction
Analytic calculation of the time evolution in two-level systems has been studied by a number of authors for a long time since the beginning years of quantum mechanics [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. These results have been applied to various areas of physics including quantum optics, laser spectroscopy, nuclear magnetic resonance and atomic collisions [15,16,17,18]. The importance of the study of quantum time-evolution is still increasing even now; For example, much attention has been paid recently to the quantum manipulation of qubits [19] and magnetization process of magnetic molecules with large spin [20]. Recent rapid development of computers has enabled massive numerical simulation of quantum dynamics. Nevertheless, it remains to be important to study analytically solvable models for the following reasons: (1) in some ranges of physical parameters the numerical simulation becomes too difficult, and (2) analytic solutions give a clearer description about parameter dependence.
Analytic solutions of quantum dynamics can be classified into several classes. Some of them are obtained by using hypergeometric functions. This was first found by Rosen and Zener [3], which has then been generalized by several authors [7,8,9,10,11,12]. In these studies, the time-variable t is generally transformed into another real variable z = z(t), which varies from 0 to 1 monotonically. Then, the Schrödinger equation of two-level systems can be reduced to the hypergeometric differential equation, and the transition probability can be related to the connection problem between two pairs of fundamental solutions around z = 0 and z = 1.
One exception is the approach by Carroll and Hioe [13]. They have studied two solvable classes, and in one of them they have introduced a new variable z(t) changing from −∞ to ∞ as t increases and have reduced the Schrödinger equation to the Riemann-Papperitz equation. Recently, Ishkhanyan has pointed out that the Carroll-Hioe model can be understood in terms of the hypergeometric functions by considering a complexvalued path z(t) = (y(t) + i)/2i where y(t) is a real variable [14]. By this complex-valued path, Ishkhanyan also found a new solvable class, but he did not obtain results for the transition probability.
In this paper, we show that for the complex-valued path, the transition probability can be calculated efficiently from the 'monodromy' matrices of the corresponding differential equations. Monodromy is one of the global properties of differential equations, and has attracted much attention by mathematicians, for example, through the deep connection with the Painlevé equations [21]. Hence, we expect that the monodromy approach is valuable not only because it enables one to calculate the transition probability for various models but also because it establishes a connection between physical phenomena and global features of differential equations.
In this paper, as a concrete example, we mainly consider the following timedependent two-level Schrödinger equation and obtain the transition probability using the monodromy associated with the solution: where the matrix elements are given by Here, the coefficients, E 0 , E 1 and V 0 , are assumed to be real constants. This is one of the solvable class reported by Ishkhanyan [14]. However, the transition probability for the model has not been obtained. It should be noted that this model is equivalent to the Rosen-Zener model [3] in the case E 1 = 0 ‡, and that it also includes the special case of the second Demkov-Kunike model [7] in the case E 0 = 0. Hence, this model can give a smooth connection between the two known results. The plan of the paper is as follows. We give the relation between the transition probability and the monodromy of the hypergeometric function in § 2, and the transition probability is calculated explicitly in § 3. The extension to the multi-level problems is ‡ In the original Rosen-Zener model, ε(t) is a constant, while V (t) has a sech-type pulse form. This Hamiltonian, however, is reformed by a proper unitary transformation of the wave function to coincide with the present model. addressed in § 4. Finally, the results are summarized in § 5. In Appendix A, we describe the generalization of the present model and its relationship to the Carroll-Hioe's model.

Hypergeometric function and monodromy
The diagonal elements in the model (1) are eliminated by the following change of variables: Then, the Schrödinger equation is expressed as By combining these two equations, we obtain the second-order differential equations for c 1 and c 2 respectively as It should be noted that the equation for c 2 is obtained by replacing ε(t) by −ε(t) in (8).
Hence, once the solution of the equation for c 1 is obtained, the solution for c 2 is easily obtained by reversing the sign of the parameters in ε(t).
The above discussion is general. Now, we consider the specific model given by (2) and (3). By substituting these specific forms of ε(t) and V (t) into (8) and adopting the change of variable as equation (8) can be reduced to the differential equation of the hypergeometric function [22], Here, the parameters, α, β and γ, are determined as In the same way, equation (9) is reduced to the hypergeometric differential equation with the parameters As already mentioned, these parameters are obtained by replacing E 0 and E 1 by −E 0 and −E 1 respectively in (12)- (14). In the following calculation, related to the calculation for c 2 , the prime indicates that it is obtained by reversing the sign of E 0 and E 1 from the original quantity without the prime. From (10), it can be easily seen that the variable |z| → ∞ as |t| → ∞. Hence, for discussion about the initial state it is convenient to consider the fundamental solutions around z = ∞ as where f ∞ (z; α) and f ∞ (z; β) are expressed in terms of the hypergeometric functions as with similar definitions for f ∞ (z; α ′ ) and f ∞ (z, β ′ ). Since α and β are pure-imaginary in the present model, we have to choose arg(z) to determine the branch. In this paper, we choose In order to decide the initial state, it is sufficient to study the limit |z| → ∞, in which case we obtain From (10), (23) and (24), A i 's and B i 's are determined.
To obtain the transition probability, we assume that the initial state is the ground state of the Hamiltonian in the limit t → −∞, It may be noted that the off-diagonal elements in (25) do not vanish. Consequently the ground state wavefunction does not correspond to |a 1 | = 1 and a 2 = 0 as it appears in the usual models. The time-evolution of the ground-state wave function is obtained generally as where The solution (26) includes an arbitary phase factor ϕ, which is chosen zero in this paper. From (26), the time-evolutions of c 1 and c 2 in the limit t → −∞ can be easily evaluated as where the phase factors are given as By using (10) and by comparing (28)-(29) with (23)-(24), the constants, A i 's and B i 's are determined as where the phase factors, ϕ 1 and ϕ ′ 1 , are given as though these are not relevant to the calculation of the transition probability. By the choice of the initial condition, the time-evolution has been described only by the fundamental solution f ∞ (z; α) around z = ∞. To be more accurate, around z = i∞ + 1/2 (corresponding to t → −∞) denoted by the point P in Fig. 1 (a), the solution is given by On the other hand, the final state is given by the solution of the hypergeometric differential equation around z = −i∞ + 1/2 (corresponding to t → ∞) denoted by the point Q in Fig. 1 (a). The path z(t) in the complex plane is also drawn in Fig. 1. Then, the solution around the point Q analytically continued from the point P does not equal to (34) and (35); The solution is expressed as linear combinations of the fundamental solutions around z = ∞. This is crucial to the calculation of the transition probability.
In order to make the situation clearer, let us deform the path of z as shown in Fig. 1 (b). In this deformed path, the analytic continuation of the solution is divided into two parts, C 1 and C 2 . Here, the path C 1 denotes a round trip to the singular point at z = 1, while the path C 2 is a half round trip around z = ∞ in the clockwise direction. The analytic continuation along the path C 2 is easily performed, and determined only by the fundamental solutions around z = ∞. On the other hand, the analytic continuation along the path C 1 is nontrivial, and determined by the global character of the differential equation called the 'monodromy'. The monodromy is expressed by the monodromy matrices as where γ(C 1 ) denotes the analytic continuation along the path C 1 . Denoting the matrix elements of Q and Q ′ as the solutions around the point Q can be expressed by Here, as shown below, the first (second) term in the final entries of (39) and (40) corresponds to the excited (ground) state in the limit t → ∞. Hence, in the calculation of the transition probability, only the element a (a ′ ) is relevant. This matrix element is calculated explicitly in the next section. Let us end this section by deriving the formula for the transition probability, using the monodromy matrix elements. In the limit t → ∞, the equations, (39) and (40), are evaluated as Here, we have suppressed the second term corresponding to the ground state. From these equations, the components of the wave function Ψ(t) = (a 1 (t), a 2 (t)) T is obtained in the limit t → ∞ as By substituting (31) and (32), the wave function Ψ(t) is evaluated as On the other hand, the Hamiltonian and the wave function of the excited state Ψ E.S. in the limit t → ∞ are given as where A and A ′ are given by (27). Then, the transition probability is calculated as where in the final equation we have introduced new variables, The last equation (47) can be used for the practical evaluation of the transition probability. The remaining task is to calculate the elements of the monodromy matrices.

Calculation of the transition probability
One may identify several ways to calculate the monodromy matrices of the hypergeometric differential equations [21]. Here, we briefly explain the simplest way.
To determine the monodromy matrix, it is crucial to use the integral representation of the hypergeometric function. By defining the integral the following relations hold: where f 0 , f 1 and f ∞ denote the fundamental solutions of the hypergeometric differential equations around z = 0, 1, ∞, respectively. Here, the constants, c pq 's, depend only on α, β and γ, and their explicit expressions are irrelevant to the present calculation. By applying the Cauchy's theorem to the integral in eq. (49), the following linear relations may be identified: where e(·) = exp(2πi·). By eliminating F 1∞ and F 0z , we obtain On the other hand, since the solution pair (F ∞0 , F 1z ) is related to the fundamental solutions around z = 1, the monodromy matrix for this pair is easily obtained as Combining (58) and (59), the monodromy matrix for (F ∞0 , F 1z ) is given as This result is easily related to the fundamental solutions, f ∞ (z; α) and f ∞ (z; β), by (52) as By comparing the above with the original monodromy matrix (36) along with (38), we finally obtain a =ã (d =d). So, as far as a is concerned, we only need to calculateR. This can be performed by straightforward but slightly lengthy calculation. As a result, we obtain the matrix element a as The matrix element a ′ for the solution c 2 is easily obtained by reversing the sign of E 0 and E 1 in the result (63). From the result for a and a ′ , the transition probability P is obtained from (47) as Let us discuss the nature of this result. The transition probability oscillates as the sech-form pulse area, πT E 0 , changes; the transition probability has minimum and maximum values as a function of E 0 as P min = sinh 2 (πT E 1 ) , for πT E 0 = nπ, , for πT E 0 = (n + 1/2)π, where n is an integer. The oscillation behavior of P for E 0 is drawn in Fig. 2 (a). The amplitude of this oscillation becomes small as E 1 increases. This feature is shown in Fig. 2 (b). In the case E 1 T ≫ max(V 0 T, 1), we obtain the ordinary Landau-Zener formula Finally, we show that the results in the limiting cases coincide with the known results. In the limit E 1 → 0, the transition probability is given as which corresponds to the Rosen-Zener formula [3]. In the limit E 0 → 0, the transition probability is given as In this case, the present model is related to the second model in Demkov and Kunike's paper [7], which corresponds to the form Their result for a = 0 corresponds to the result (68).

Application of monodromy to multi-level problems
The application of the monodromy matrix to the transition probability is not restricted to the hypergeometric functions. The monodromy approach is also applicable to the differential equations whose monodromy is known. To show such an example, we consider the multi-level problem. We expect that more solvable classes can be found by using the present approach.
In this section, we treat the following time-dependent Hamiltonian: where the time-dependent part ε(t) is given as and V j 's (2 ≤ j ≤ N) are constants. It should be noted that in the limit E 1 T → ∞, this model is reduced to the extended Landau-Zener model studied by several authors [23,24,25,26]. To eliminate the diagonal element of the Hamiltonian the wave function denoted by Ψ(t) = (a 1 , a 2 , · · · , a N ) T is transformed into new variables as The integral in the exponent is then calculated as Thus, the Schrödinger equation is obtained as where By changing the time variable as z = sinh(t/T ), the equations are modified as We make a further change of variables as where λ j 's are arbitrary constants satisfying N j=2 λ j = 1.
Consequently, we finally obtain and for 2 ≤ i ≤ N This is the Okubo equation expressed by where I is the identity operator, C is a diagonal matrix, and A is a general matrix. This equation has been studied by Okubo in detail [27], and it is known that this form of equation is convenient to study the monodromy. Thus, it has been shown that at least one specific model of multi-level systems can be reduced to the differential equation whose monodromy is known. Actual calculation of the transition probability needs explicit treatment of the monodromy matrices, and remains as a future problem. The present discussion for multi-level systems is preliminary, and more detailed study will be needed to clarify the efficiency of the monodromy approach.

Summary
We have calculated the transition probability for the Hamiltonian including the tanhtype plus sech-type energy difference with constant off-diagonal elements. The obtained result gives the natural connection between the known results, the Rosen-Zener model and the second Demkov-Kunike model. This model also includes the Landau-Zener formula in the limit of the large amplitude of the tanh-type energy difference.
In our calculation, the monodromy of the hypergeometric functions is essential. We have shown that the monodromy approach is also applicable to the multi-level problems. We expect that the use of the monodromy in the calculation of the transition probability does not only helps finding more solvable models but also connects global properties of the differential equation with the physical phenomena. Details of calculation especially for the multi-level problem remain as future problems.