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Zener dynamics beyond Zener's assumptions
Physica A 168 (1990) 456-468 North-Holland
ZENER DYNAMICS BEYOND ZENER'S ASSUMPTIONS D r o r L U B I N a, Yuval G E F E N a a n d I s a a c G O L D H I R S C H b ~Department of Nuclear Physics, The Weizmann Institute of Science, Rehovot 76 100, Israel bDepartment of Fluid Mechanics and Heat Transfer, Faculty of Engineering, Tel Aviv University, Tel Aviv 69 978, Israel Zener's treatment of the tunneling process between two crossing diabatic levels is generalized to include time effects, as well as time dependence of the participating levels and their intercoupling. All of these can affect in a significant way the various transition probabilities of interest. These generalizations are necessary if the Zener process is to serve a useful role in describing transitions in N-level systems. We find a possible nonmonotonic dependence of the transition probability on the rate of change of the external bias (e.g. a magnetic field). Power law corrections to the leading transition probability are shown to exist when finite time effects are included. Finally we wish to mention the possible saturation of the localization length in energy space as a function of the bias in a many-level system in the sudden limit.
I. Introduction T h e s t u d y of d y n a m i c s in m e s o s c a l e s y s t e m s has r e c e n t l y e n j o y e d i n c r e a s i n g a t t e n t i o n . O n e o f the basic m e c h a n i s m s for e l e c t o r n i c t r a n s i t i o n s in this c o n t e x t is that o f Z e n e r t u n n e l i n g , t h e e s s e n t i a l p r o p e r t i e s of which h a v e b e e n e l u c i d a t e d in t h e s e m i n a l p a p e r by C. Z e n e r [1]. W h i l e this classic w o r k is fairly c o m p l e t e u n d e r Z e n e r ' s s i m p l i f y i n g a s s u m p t i o n s , o n e m a y i n q u i r e into t h e a p p l i c a b i l i t y o f his results to cases w h e r e s o m e o f t h e s e a s s u m p t i o n s h a v e to be r e l a x e d . In fact, w h e n Z e n e r ' s t h e o r y is e m p l o y e d to a n a l y z e d y n a m i c s of small q u a n t u m s y s t e m s , a n d in p a r t i c u l a r w h e n c o n s e c u t i v e Z e n e r t r a n s i t i o n s a r e i n v o l v e d , o n e is r e q u i r e d to go b e y o n d t h e original analysis of a t w o - l e v e l system. In t h e p r e s e n t p a p e r we f o r m u l a t e a n d a n a l y z e a few e x t e n s i o n s to Z e n e r ' s t h e o r y . W e show t h a t r e l a x i n g Z e n e r ' s a s s u m p t i o n s m a y l e a d to significant c h a n g e s in the t r a n s i t i o n p r o b a b i l i t i e s . O u r results, o b t a i n e d for s e v e r a l k i n d s o f t w o - l e v e l c o n f i g u r a t i o n s , a r e t h e n a p p l i e d to the case of m u l t i - l e v e l s p e c t r a , c h a r a c t e r i s t i c of m e s o s c o p i c q u a n t u m systems. In p a r t i c u l a r , it was s h o w n that u n d e r fairly g e n e r a l c o n d i t i o n s t h e r e is l o c a l i z a t i o n in t h e e n e r g y s p a c e o f such m a n y - l e v e l systems [2, 3]. O u r results i n d i c a t e t h a t t h e l o c a l i z a t i o n l e n g t h , ~:, d o e s n o t d i v e r g e with t h e r a t e o f c h a n g e o f t h e e x t e r n a l p a r a m e t e r ( i . e . , with 0378-4371/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)
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the external bias), in accordance with our previous results [4], pertaining to one-dimensional N-level systems. The structure of the present paper is as follows. Section 2 provides a brief description of the gneeral two-level Zener problem. The specific assumptions involved in the reduction to Zener's original formulation are exposed. In section 3 we discuss certain extensions of the traditional analysis. First, for the original Hamiltonian, we evaluate the transition probability at finite times, which evidently differ from the asymptotic expressions. One source of these modifications of the asymptotic transition probabilities is that at finite times the adiabatic states differ from the diabatic states by a time dependent transformation. Another correction results from the finite time correction terms in Zener's original analysis. We next consider modifications related to the explict time dependence of the diabatic states (still with a 2 × 2 matrix of the form discussed by Zener). Finally we consider the implications of time dependent off-diagonal terms in the Hamiltonian. In section 4 we summarize the results, and discuss the applicability of our analysis to the description of dynamics in many level systems. We formulate conditions to approximate such dynamics by consecutive (two-level) Zener transitions, and evaluate how the modifications alluded to above affect the localization length in energy space.
2. The Zener problem In the present section we present a formulation of the Zener problem in a way which is convenient for our further analysis. Consider a system having two quantum levels, which are allowed to depend on an externally controllable parameter s. Such a parameter could be, for instance, an external time dependent magnetic field. For each value of the parameter s we consider two diabatic states, X~(S) and )(2(S), which are eigenstates of a Hamiltonian Ho(s ). Their corresponding eigenvalues are denoted by El(s ) and/~2(s), respectively. In addition, it is assumed that the above Hamiltonian H 0 is only a certain first order approximation. The two levels are actually coupled and the "correct" Hamiltonian is H~(s) = Ho(s ) + ~H(s). Staying within the two-level world we can thus write
Ha(S) X1(S) =
(la)
E,(S) Xl(S) + a(S) X2(S) ,
x (s) = E (s) X (s) +
a*(s) X,(s)
,
(lb)
where A(s)=--(X2(s),~H(s)xl(s)) denotes the coupling created by ~H(s) (( f, g) is the scalar product of f and g), the diagonal part of 8H is absorbed in the redefinition of the energies: Ei(s ) =-- ff~i(s) + ( Xi(s), ~H(s) X~(S) ) , and her-
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miticity of Ha(S ) is manifest. Following Zener's work we shall assume that E l(s) = E 2(s) for some value of s, and E I ¢ E 2 for other values of s. Without loss of generality, one can fix s = 0 to be the crossing point: E~(0) = E:(0). The corresponding time dependent problem, which is of interest when s is varied, can then be described by considering any initial state which is a superposition of XI and X2 at the appropriate (initial) time. The pertinent equation reads 0 ih ~ [a(s) X~(s) + b(s) X2(s)] = Ha(s ) [a(s) X1 (s) + b(s) X2(s)],
(2)
where a, b are the probability amplitudes in the corresponding states. Using eq. (1) we obtain ih~(~
X,(s) + a(s)
dxi(s)+ ~db(s) X2(s) + b(s) dx2(s) ] ds /
= a(s) [E,(s) X,(S) + A(s) Xz(S)] + b(s) [E2(s ) gz(s) + Zl*(s) g,(s)].
(3)
Upon projecting eq. (3) on the (orthogonal) states X~ and X2, we obtain
ihg[a'(s) + ( X,(S), X'l(s) )a(s) + ( X~(S), X'z(s) ) b(s)] (4a)
= E,(s) a(s) + a*(s) b(s), ihg[b'(s) + (X2(S), g'2(s))b(s) + (X2(s), X'~(s))a(s)]
(4b)
= E2(s ) b(s) + A*(s) a(s),
where a prime denotes differentiation with respect to s. Note that (Xi(S), X'i(s)) is a pure imaginary quantity, and that the eigenstates Xl, X2 are defined up to an arbitrary phase (which can be chosen to be s dependent). It is easy to show that one can choose these phases to satisfy ihd ( X, (s), X'I (s)) = Ez(s ) ,
(5a)
ih~ (X2(s), X'2(s)) = E,(s).
(5b)
Indeed this choice is made below. It is convenient to rewrite eqs. (4) in matrix notation,
ihg ds \b(s)
=
a(s)-ihi(x2, x])
with E,2(s ) =- E, (s) - E2(s ).
-E,2(s)
b(s)]'
(6)
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D. Lubin et al. / Zener dynamics beyond Zener's assumptions
The matrix in eq. (6) can be considered as a "dynamical Hamiltonian", H d ( s ) , which in general differs from the "adiabatic Hamiltonian" of the problem, Ha(s ). The off-diagonal term of H d (e.g. the (1, 2) element) can be regarded as an effective coupling z~(s), which may be represented as A ( s ) = d(s) exp[i0(s)], where d and 0 are real functions of s. It is convenient to define the following variables: Y(s) ~ a(s) e i0(,)/2 ,
1)(s) =- b(s) e +i°(')/2 .
(7)
Clearly lal 2 = lal 2 and I/~[2 = ]bl 2, i.e. the probabilities to be in either state are independent of this change in phase. The resulting equations for if(s) and/~(s) read
d ihg
=
d(s, g)
-(E
z(s ) + ½hgO'(s, g))
"
(8)
Note that in the (tT,/~) representation the dynamical Hamiltonian matrix, H d, is real. In the general case the elements of H d may depend explicitly on g. As there are no approximations involved in the derivation of eq. (8) from the assumed two-level dynamics, it constitutes a general formualtion of the twolevel problem. In the remainder of this section we describe Zener's approach [1] to the solution of the dynamical problem presented by eqs. (6) or (8). The physical assumptions are the following. The crossing point is sharply defined, that is E12(s ) is large, except in the close vicinity of s = 0. The coupling term zl(s) is small compared with E~2(s ) outside the above-mentioned region. All the quantities in the problem, other than E12 , have a very slow s dependence, hence their derivatives with respect to s may be neglected. Quantitatively, this means that if we define a scale of s, denoted by Sze ., in which most of the transition process takes place, then we assume f ' ( 0 ) Sze n ~ f(0), where f is e.g. the magnitude of an off diagonal element of Hd; g is replaced by a constant (denoted by a). When s is an externally controlled parameter the latter is not necessarily an assumption but rather a specification of the problem considered. When all the above assumptions hold, H d is approximately diagonal, except in a small vicinity of s = 0. It can be assumed that the transition probability is determined by the form of H d near s = 0, while in other regions of s one may replace H 0 by another matrix which keeps the states virtually uncoupled. Specifically it is assumed that for every s we may approximate the elements of H d by the first non-vanishing term in their expansion around s = 0. Thus El2(S ) may be approximated by El2(0 ) s =-/xs, and zl(s) by a _= A(0). (The latter is valid provided that we can neglect the overlap terms (X1, X'2), (X2, X'~), as
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D. Lubin et al. / Zener dynamics beyond Zener's assumptions
well as 0'.) Having invoked all these approximations and assumptions, eq. (8) simplifies to
i h a -~s \ b(s)
(9)
b(s) / "
Eq. (9) is the model solved by Z e n e r in actuality. Since all transitions are assumed to occur in a small neighborhood of s = 07 one can find the various transition amplitudes by considering a p r e p a r e d state at s = - ~ evolving according to eq. (9) and inquire into the nature of the corresponding state at s = ~. When all of the above assumptions are justified, the answer obtained corresponds to the transition from a given state at s < 0 just outside the transition region to a state at s > 0 slightly beyond that region. As is well known (and easily seen from the matrix in eq. (9) or that in eq. (8)) the (adiabatic) eigenstates of the o p e r a t o r H are nondegenerate. For s < 0 and far enough from the transition region, the adiabatic states are basically the same as the diabatic ones; a similar statement holds for s > 0 . However, following the crossing point the diabatic state with the lower energy becomes the one with the higher energy. The higher energy adiabatic eigenstate corresponds to X2(s) (say) for s---~ - ~ and to X~(S) for s---~ + ~ and vice versa. Therefore a transition from one adiabatic state to the other means (for large [sl) staying on the same diabatic state. It is customary to define the transition amplitude between X~ and X~ as t (standing for transmission), between X2 and X2 as t', between X~ and X2 as r (reflection), and betwen X2 and X~ as r'. The transition matrix between the adiabatic levels is therefore
Clearly U is a unitary matrix, which must obey the unitary conditions Irl 2 = Ir'l 2
Itl ~ = It'l 2
Irl 2 + Itl 2 = 1
tr* = - t ' * r '
(11)
In order to obtain more detailed relations, we should go back to eq. (9) and note that it is invariant when J, b are replaced by /~*, - J * respectively, and the complex conjugate is taken. It follows that t'= t*,
r'= -r*.
(12)
The result obtained by Z e n e r for the problem formulated in eq. (9) was
It] 2 =
e-~,
A2 T -= hp~ol "
(13)
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461
The result in eq. (13) is an exact consequence of eq. (9) when initial and final conditions are specified at --- oo. In the next section we discuss how the result given by eq. (13) is modified when the conditions are different.
3. Going beyond Zener's analysis In the previous section we have set the ground for the analysis of a two-level system. We are now in a position to deviate from Z e n e r ' s original analysis, and study the ensuing modifications of the transition probabilities. Several scenarios of such modifications are discussed below. We first consider a finite integration range, and then corrections due to the s dependence of H d. 3.1.
Transition p r o b a b i l i t i e s at f i n i t e times
In this section we consider the Hamiltonian matrix of eq. (9), which is Z e n e r ' s model, but evaluate the transition probability over a finite time interval (for brevity, we call s " t i m e " in this section). The set of equations (9) was solved by Z e n e r [1]. The result in eq. (13) applies when the limits of integration are taken from s = - ~ to s = + ~. We now calculate how the result is affected when the integration is taken from s = - s 2 to s = + s I, keeping all other assumptions intact. To simplify the calculations we shall consider only finite s 1, letting - s 2 remain infinitely large. The extension to the case where s 2 is finite as well is staightforward, but the expressions obtained are m o r e complicated. There are two sources for corrections when finite times are considered. One arises from the necessity to consider the actual solution of the pertinent equations at a finite time (in this case, the finite time value of the W e b e r function, cf. ref. [1]). The second correction arises from the fact that Z e n e r ' s calculation is p e r f o r m e d in the diabatic basis. In the large Isl limit the diabatic and adiabatic states of Z e n e r ' s problem coincide. For finite times, however, the transformation from the diabatic basis to the adiabatic eigenstates should be taken into account. The first problem would be to calculate the corrections to t, r when finite times are considered. Recently, M/illen et al. [5] have estimated the time, denoted in section 2 by Sze ., required for the transition to be completed. They considered the probability of staying on the same diabatic level I (x)l 2 as a function of the dimensionless time x =-- t x s / A (Xl --= t z s l / A , Xzo, =- lZSze,/A). Starting with a value lal 2 = 1 when x is a large negative number, the evolution of lal 2 towards its x---,oo asymptotics was studied. The width of the decay region was used as the definition of Xze .. This width was obtained using the
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D. Lubin et al. / Zener dynamics beyond Zener's assumptions
scaling properties of this profile. Their results, transalated to our notation, are that Szcn = ~ (equivalently Xz~n = 1 / x / ~ ) in the sudden limit (Y ~ 1, Y was defined in eq. (13)), and SZe~ = A/k~ (Xz~" = 1) in the adiabatic limit (T~I)
#l.
Here we analyze this problem further. Following Zener, the solution of eq. (9) can be expressed in terms of Weber functions. The initial and final values of ItT(x)l2 are then determined from the asymptotic behavior of the Weber functions. In practice, the Weber function reaches is asymptotic values (corresponding to x---> -+~) to a good approximation for finite values of x, and the transition between its asymptotic values occurs in a narrow range of values of x. The value of Xz~. can be defined as the width of the region in which most of the transition takes place. Using the asymptotic analysis of the parabolic cylinder functions [7], we find that when 3' is of order unity or smaller the crossover to asymptotic behavior is when x ~ 1/x/-y, while for large -/it occurs at x = 1. We conclude that the results for the transition time obtained by both approaches agree. We now assume that indeed x 1 is large enough so that we are in the asymptotic regime, and compute the leading corrections to the transition probability. In that case we may follow Zener's original analysis, up to the last step, where we compute not only the asymptotic term, but also the leading correction to it [8]. We obtain
1~7[2 = e - ~
1+
1
1
+4x~
V ~ e-3~,/4 Re(eXp{i[27x~ + 87 l n ( x / v ~ ) + 7]/4} ) xl r * ( - i ~ , / 2 + 1) .
(14)
The general form of [ff(x)l 2 therefore consists of a sum of a slowly varying (monotonously decreasing) function, approaching the asymptotic Z e n e r value, and a fast oscillating term, whose magnitude decays to zero. Consider again the limiting cases: In the sudden limit (Y ~ 1) the amplitude of the oscillatory correction term is proportional to V~/x. At the crossover to the asymptotic regime (x = Xzcn) the magnitude of the oscillatory part of ItT[2 is comparable to 1/~[2 ~'~-'Ti"yand should not be neglected. In the adiabatic limit ( y >>1), the amplitude of the oscillations is proportional to e x p ( - ' r r 7 / 2 ) / x . In this limit, however, the term 1/4x~ dominates the results up to exponentially large times (x I >> exp('rry/2)). For intermediate values of 3' the amplitude of the oscillations at the crossover is of order unity. We conclude therefore that if y ~< 1 the ~ Defining a "superadiabatic'" basis, M.V. Berry has recently shown that the transition time in the slow rate limit is given by hX/'ff~ [6].
D. Lubin et al. / Zener dynamics beyond Zener's assumptions
463
oscillations of [~l z are significant at x = Xze.. If y is much larger than unity, the asymptotic transition probability is not yet attained at x = Xzen. The second source of corrections to Zener's formula is due to the transformation between the diabatic and adiabatic bases. The integration of eq. (9) (or the approximate solution eq. (14)) yields the (time dependent) transition coefficients (in the diabatic basis) t ~ Ill exp(i0,) and r ~ Irl exp(i0r). Earlier (cf. the analysis at the end of section 2) we have defined the transition coefficients at asymptotically large times. At finite times one has to distinguish between transition coefficients in the scheme of diabatic states, and those in the scheme of adiabatic states. We note that there is a non-trivial, time dependent, transformation between the schemes. We denote the corresponding transmission coefficient in the adiabatic representation by t a. The time dependent tansformation between the diabatic to the adiabatic bases is readily calculated by diagonalizing the Hamiltonian in eq. (9) (note that in Zener's approximation H d = Ha). In order to obtain corrections to the same order as those previously discussed, we expand the transformation matrix in powers of 1 / x 1. The result is
( 1)
Ita[2=It[ 2 1--~X~
1
Irlltl
+ 4x~-- - - x l cos(O, - Or).
(15)
(Note that the phases 0t, 0r increase with Xl, therefore the last correction term is oscillatory.) We obtain therefore a very similar result to the previous correction discussed: for ranges of values of x beyond the crossover value ( X z e n ) , ]ta]2 has a correction to its asymptotic value that decays like 1/x 2, and another oscillatory term which decays like 1 / x . The former correction is of the same order of magnitude as the one found before (cf. eq. (14)), but with an opposite sign. The amplitude of the oscillatory term has the same y dependence as the amplitude of the oscillatory term in eq. (14). These two types of corrections are independent of each other, and the complete result for the transition probability is obtained by substituting 1~1 from eq. (14) for It[ in eq.
(15). Note that in the extreme sudden limit ( y - - ~ ) , when Irl =0, there are no oscillations, but there are still corrections (of both types) of order x -2 to
Itl: l. Estimating Xzen becomes especially important when we study a system whose spectrum contains more than one narrow gap. In this case it is important to know under what conditions can the large system be approximated by a set of independent, decoupled, two-level Zener transitions. A necessary condition is obviously that the time that elapses between consecutive transitions is much larger than the time required to complete the transition. Our results indicate
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D. Lubin et al. / Zener dynamics beyond Zener's assumptions
that the latter time is not necessarily given by Xze. discussed above, but may be much longer. At x = Xzen there are significant corrections to the asymptotic result. For larger times the corrections decay like a power law of time.
3.2. The role of the dynamical Hamiltonian In this section we analyze how eq. (13) is modified when not all the assumptions leading to eq. (9) are valid. The first assumption we shall relax is that the overlap term c(s)=-(Xl(s), X'2(s)) may be neglected. Indeed, unless c(s) vanishes exactly due to some symmetry of Ho(s), it is not necessarily a small quantity for any value of s. When c(s)~O, ~l(s) in eq. (6), and consequently d(s) and O(s) in eq. (8), may depend explicitly on a. In particular, for large a, d may become large, and the transition region is increased, rendering the lowest order expansion invalid. The general problem is thus rather complicated. It may still be possible, however, to employ Zener's approach even in this case, provided the value of a is not much larger than a c =- [A(s)[/hlc(s)l. When ~ satisfies this condition, d(s) is of the same order of magnitude as A(s) (or smaller), and the transition region does not increase. In that case we may still expand H j to lowest orders around the crossing point of the effective diabatic levels, which in general will be shifted from s = 0 (see eq. (8)). To illustrate the possible consequences of this modification we consider a simple example: assume that c(s) is nearly constant, that is while we may not neglect the derivatives of Xi(s), we may do so for c(s). As a result 0 is constant, and H~ is expanded around s = 0 for every a. In that case we end up with a matrix of a similar form to the one solved by Zener, and his solution applies to this problem too. The crucial difference is that the effective coupling in the definition of y in eq. (13) is now = ,1(0) = A(0) - io~hc(0).
(16)
In this example we consider again asymptotically large times. Following Zener's analysis, the asymptotic transition probability, It[ 2, is
Itl 2 = e x p [ -
~ \(IA]2 ha
_2im(Ac,)+ha[c]2)]
is no longer monotonous in a. In of a: a = a c. In this case
fact
It] 2 has a maximal value for
It]max = exp - ~ - [IA]lc[ - Im(Ac*)]
.
(17)
afinite
value
(is)
D. Lubin et al. / Zener dynamics beyond Zener's assumptions
465
It is important to note that the value of a for which the non-monotonous behavior is obtained is in the regime of values of a for which the approximations used are valid. The condition for Itlmax 2 = 1 is C[A I = ialcl. Unlike in the original Z e n e r formula, in which It[--> 1 as a - - ~ , the corrected formula predicts Itl--~0 in this limit. In summary, the transition probability Itl 2 increases from It[ = 0 at a = 0 to 2 ]/tmax a t a c, and decreases for a > a c to Itl = 0 at a = ~ . For a >>a c the approximation breaks down. The above result is exact only for constant A and c. However, as long as the variation of A, c with s is not too rapid, changing a will shift the crossing point slightly. This will modify the effective value of y, but the same qualitative behavior will be observed. As another simple example, we consider the case where indeed c(s) =- O, but = A does depend on s. To make things even simpler, yet illustrative, we demonstate that such an s dependence may have profound consequences by considering a specific family of functions ,~(s), which is convenient for analysis, i.e. ,~(s)= d exp[i0(s)], where d is a real constant, and only the phase 0 is s-dependent. All the systems in this family have exactly the same adiabatic spectrum; therefore Zener's assumptions are of equal validity for every one of them. If a is the same in every case, the Z e n e r formula predicts exactly the same transmission probability for all members of this family. We shall see, however, that in spite of this similarity, the s dependence of 0 causes a non-universal behavior of the transition probabilities. Consider now eq. (8) with constant d and O(s). One obvious implication of eq. (8) is that a nonzero value of O'(s -- 0) produces a finite shift in the crossing point of the effective diabatic levels. For some choices of O(s) there may be more than one crossing point, or none at all. The value of the slope It depends of course on the position of the crossing. When one takes E~2(s ) = Its as an exact relation valid for every s and a constant O'(s) = [3, the crossing point is shifted from s = 0 to s = ha/3/2it, and nothing else changes. In this case the result for the transition probability is identical to that of Zener's, up to a shift of the time in which the transition occurs. If one keeps the assumption that E12(s ) = Its, but now assumes 0 ' = 2/3s, the diagonal terms in eq. (2)) read: +-(It +ha/3)s. Consequently, in this case, Z e n e r transition probability is obtained by replacing It in eq. (13) by (It + ha~3):
( Itl 2
= exp
d2 -or ah(t t + ha/3)
) "
(19)
Hence, the value o f / 3 affects significantly the transition probability. We conclude this section by reiterating that the properties of Z e n e r transitions are not determined only by the adiabatic spectrum and the driving rate: they are strongly influenced by the nature of the off-diagonal coupling term [9].
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D. Lubin et al. / Zener dynamics beyond Zener's assumptions
4. Discussion
In this work we have analyzed in detail the Zener problem of transitions between two adiabatic energy levels, and found corrections and deviations from Zener's analysis. We have shown that when finite times are considered there are significant corrections to the transition probability. When the s dependence of the various elements in the system is taken into account, significant corrections may arise. By considering the derivatives of the diabatic states and keeping all the other assumptions intact, we obtain a qualitative change in the behavior of the transmission probability: it is nonmonotonous in the driving rate. Finally we have shown that two-level systems with identical spectrum, driven at the same rate, can show significantly different behavior, depending on the details of the system. The conclusion here is that the validity of the Z e n e r assumptions cannot be determined merely from the structure of the adiabatic spectrum, but one has to consider the full details of the specific system at hand. One of the important applications of the two-level calculation is to the analysis of a more realistic many-level quantum system. The reduction of the N-level problem to a set of many two-level problems follows two assumptions. The first is transitions from a given level to levels not adjacent to it are neglected, and practically we have only two-level couplings. The second assumption, which is pertinent only when the first one is satisfied, is that these two level systems are uncoupled, and each is described by Zener's dynamics. In this paper we have concentrated on the second problem: the corrections and deviations from the two-level analysis, assuming that the first assumption holds. Recently we have analyzed the problem of a metallic ring threaded by a time dependent magnetic flux [4]. In that problem there is a many-level adiabatic spectrum (see fig. lb), the magnetic flux plays the role of the parameter s, and there are transitions among the adiabatic levels. The result of solving exactly the dynamical problem is that localization in energy space is obtained. We have found a f i n i t e localization length, consistent with transmission probabilities less than unity, even for infinitely large a. This result was obtained directly in the N-level system, and was in contradiction with the result obtained when the same system was approximated by a set of separable Z e n e r transitions. One possible source of this contradiction is the above discussed finite time effect. The integration limit is necessarily finite, and, as we have shown, finite size corrections decay only quadratically. Even in the extreme sudden limit, this will induce nonnegligible deviation from the value Itl = 1 at each transition. We note that in the ringe problem, when two diabatic states Xi, Xj cross as a function of s, the symmetry of the Hamiltonian results in c~-(Xi, X'i)-=0;
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)l
I ....
I ....
-0.5
-i
ibl
?x
+ , , i
I
i i i ,
0
i
I ....
0.5
i , i ,
I
, i , i
I
3..0
0.5
"~
0.0
t.t_l
-0.5
-3..0 ,il
....
-i
I . . . .
-0.5
I ....
0
I ....
0.5
I
1
Fig. 1. (a) The two-level spectrum: two linear diabatic energy levels (a, b) which cross (dashed lines), and the resulting adabatic spectrum (up, down) when coupling is introduced (solid lines). (b) An arbitrary N = 9 adiabatic level spectrum with narrow gaps. Specific systems may have a more complicated spectrum. The circles of radius Szo" (defined in the text) illustrate the approximation of separating the N-level system into a collection of isolated two-level systems.
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D. Lubin et al. / Zener dynamics beyond Zener's assumptions
t h e r e f o r e we do n o t e x p e c t , n o r o b s e r v e , n o n m o n o t o n i c i t y o f the l o c a l i z a t i o n l e n g t h as a f u n c t i o n of c~. In c o n c l u s i o n , e v e n w h e n t r a n s i t i o n s in a m u l t i - l e v e l s y s t e m can b e a p p r o x i m a t e d b y c o n s e c u t i v e Z e n e r t r a n s i t i o n s , t h e l a t t e r s h o u l d b e c a l c u l a t e d with p r o p e r care: finite t i m e effects a n d the influence o f the r a t e of c h a n g e o f the e x t e r n a l bias, as well as t h e t i m e d e p e n d e n c e o f scalar p r o d u c t s o f w a v d f u n c t i o n s , s h o u l d be i n c l u d e d . In p r i n c i p l e o n e s h o u l d also c o n s i d e r c o r r e c t i o n s d u e to c o u p l i n g s a m o n g n o n a d j a c e n t levels.
Acknowledgements This w o r k was s u p p o r t e d in p a r t b y t h e U . S . - I s r a e l B i n a t i o n a l Science F u n d ( B S F ) , a n d the M i n e r v a F o u n d a t i o n , M u n i c h , F e d . R e p . G e r m a n y .
References [1] C. Zener, Proc. R. Soc. London A 137 (1932) 696; L. Landau, Soy. Phys. 1 (1932) 89; E.G.C. Stueckelberg, Helv. Phys. Acta 5 (1932) 369. [2] Y. Gefen and D.J. Thouless, Phys. Rev. Lett. 59 (1987) 1752. [3] G. Blatter and D.A. Browne, Phys. Rev. B 37 (1988) 3856. [4] D. Lubin, Y. Gefen and I. Goldhirsch, Phys. Rev. B 41 (1990) 4441. [5] K. Miillen, E. Ben-Jacob, Y. Gefen and Z. Schuss, Phys. Rev. Lett. 62 (1989) 2543. [6] M.V. Berry, Histories of Adiabatic Quantum Transitions, University of Bristol preprint (1989). [7] See e.g.: M. Abramowitz and I.A. Stegun, eds. Handbook of Mathematical Functions, p. 685. [8] See e.g.: E.T. Whittaker and G.N. Watson, A Course of Modern Analysis (Cambridge Univ. Press, Cambridge, 1952) p. 348. [9] S. Fishman, K. M/illen and E. Ben-Jacob, Phys. Rev. A, to be published.
ZENER DYNAMICS BEYOND ZENER'S ASSUMPTIONS D r o r L U B I N a, Yuval G E F E N a a n d I s a a c G O L D H I R S C H b ~Department of Nuclear Physics, The Weizmann Institute of Science, Rehovot 76 100, Israel bDepartment of Fluid Mechanics and Heat Transfer, Faculty of Engineering, Tel Aviv University, Tel Aviv 69 978, Israel Zener's treatment of the tunneling process between two crossing diabatic levels is generalized to include time effects, as well as time dependence of the participating levels and their intercoupling. All of these can affect in a significant way the various transition probabilities of interest. These generalizations are necessary if the Zener process is to serve a useful role in describing transitions in N-level systems. We find a possible nonmonotonic dependence of the transition probability on the rate of change of the external bias (e.g. a magnetic field). Power law corrections to the leading transition probability are shown to exist when finite time effects are included. Finally we wish to mention the possible saturation of the localization length in energy space as a function of the bias in a many-level system in the sudden limit.
I. Introduction T h e s t u d y of d y n a m i c s in m e s o s c a l e s y s t e m s has r e c e n t l y e n j o y e d i n c r e a s i n g a t t e n t i o n . O n e o f the basic m e c h a n i s m s for e l e c t o r n i c t r a n s i t i o n s in this c o n t e x t is that o f Z e n e r t u n n e l i n g , t h e e s s e n t i a l p r o p e r t i e s of which h a v e b e e n e l u c i d a t e d in t h e s e m i n a l p a p e r by C. Z e n e r [1]. W h i l e this classic w o r k is fairly c o m p l e t e u n d e r Z e n e r ' s s i m p l i f y i n g a s s u m p t i o n s , o n e m a y i n q u i r e into t h e a p p l i c a b i l i t y o f his results to cases w h e r e s o m e o f t h e s e a s s u m p t i o n s h a v e to be r e l a x e d . In fact, w h e n Z e n e r ' s t h e o r y is e m p l o y e d to a n a l y z e d y n a m i c s of small q u a n t u m s y s t e m s , a n d in p a r t i c u l a r w h e n c o n s e c u t i v e Z e n e r t r a n s i t i o n s a r e i n v o l v e d , o n e is r e q u i r e d to go b e y o n d t h e original analysis of a t w o - l e v e l system. In t h e p r e s e n t p a p e r we f o r m u l a t e a n d a n a l y z e a few e x t e n s i o n s to Z e n e r ' s t h e o r y . W e show t h a t r e l a x i n g Z e n e r ' s a s s u m p t i o n s m a y l e a d to significant c h a n g e s in the t r a n s i t i o n p r o b a b i l i t i e s . O u r results, o b t a i n e d for s e v e r a l k i n d s o f t w o - l e v e l c o n f i g u r a t i o n s , a r e t h e n a p p l i e d to the case of m u l t i - l e v e l s p e c t r a , c h a r a c t e r i s t i c of m e s o s c o p i c q u a n t u m systems. In p a r t i c u l a r , it was s h o w n that u n d e r fairly g e n e r a l c o n d i t i o n s t h e r e is l o c a l i z a t i o n in t h e e n e r g y s p a c e o f such m a n y - l e v e l systems [2, 3]. O u r results i n d i c a t e t h a t t h e l o c a l i z a t i o n l e n g t h , ~:, d o e s n o t d i v e r g e with t h e r a t e o f c h a n g e o f t h e e x t e r n a l p a r a m e t e r ( i . e . , with 0378-4371/90/$03.50 © 1990- Elsevier Science Publishers B.V. (North-Holland)
D. Lubin et al. / Zener dynamics beyond Zener's assumptions
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the external bias), in accordance with our previous results [4], pertaining to one-dimensional N-level systems. The structure of the present paper is as follows. Section 2 provides a brief description of the gneeral two-level Zener problem. The specific assumptions involved in the reduction to Zener's original formulation are exposed. In section 3 we discuss certain extensions of the traditional analysis. First, for the original Hamiltonian, we evaluate the transition probability at finite times, which evidently differ from the asymptotic expressions. One source of these modifications of the asymptotic transition probabilities is that at finite times the adiabatic states differ from the diabatic states by a time dependent transformation. Another correction results from the finite time correction terms in Zener's original analysis. We next consider modifications related to the explict time dependence of the diabatic states (still with a 2 × 2 matrix of the form discussed by Zener). Finally we consider the implications of time dependent off-diagonal terms in the Hamiltonian. In section 4 we summarize the results, and discuss the applicability of our analysis to the description of dynamics in many level systems. We formulate conditions to approximate such dynamics by consecutive (two-level) Zener transitions, and evaluate how the modifications alluded to above affect the localization length in energy space.
2. The Zener problem In the present section we present a formulation of the Zener problem in a way which is convenient for our further analysis. Consider a system having two quantum levels, which are allowed to depend on an externally controllable parameter s. Such a parameter could be, for instance, an external time dependent magnetic field. For each value of the parameter s we consider two diabatic states, X~(S) and )(2(S), which are eigenstates of a Hamiltonian Ho(s ). Their corresponding eigenvalues are denoted by El(s ) and/~2(s), respectively. In addition, it is assumed that the above Hamiltonian H 0 is only a certain first order approximation. The two levels are actually coupled and the "correct" Hamiltonian is H~(s) = Ho(s ) + ~H(s). Staying within the two-level world we can thus write
Ha(S) X1(S) =
(la)
E,(S) Xl(S) + a(S) X2(S) ,
x (s) = E (s) X (s) +
a*(s) X,(s)
,
(lb)
where A(s)=--(X2(s),~H(s)xl(s)) denotes the coupling created by ~H(s) (( f, g) is the scalar product of f and g), the diagonal part of 8H is absorbed in the redefinition of the energies: Ei(s ) =-- ff~i(s) + ( Xi(s), ~H(s) X~(S) ) , and her-
D. Lubin et al. / Zener dynamics beyond Zener's assumptions
458
miticity of Ha(S ) is manifest. Following Zener's work we shall assume that E l(s) = E 2(s) for some value of s, and E I ¢ E 2 for other values of s. Without loss of generality, one can fix s = 0 to be the crossing point: E~(0) = E:(0). The corresponding time dependent problem, which is of interest when s is varied, can then be described by considering any initial state which is a superposition of XI and X2 at the appropriate (initial) time. The pertinent equation reads 0 ih ~ [a(s) X~(s) + b(s) X2(s)] = Ha(s ) [a(s) X1 (s) + b(s) X2(s)],
(2)
where a, b are the probability amplitudes in the corresponding states. Using eq. (1) we obtain ih~(~
X,(s) + a(s)
dxi(s)+ ~db(s) X2(s) + b(s) dx2(s) ] ds /
= a(s) [E,(s) X,(S) + A(s) Xz(S)] + b(s) [E2(s ) gz(s) + Zl*(s) g,(s)].
(3)
Upon projecting eq. (3) on the (orthogonal) states X~ and X2, we obtain
ihg[a'(s) + ( X,(S), X'l(s) )a(s) + ( X~(S), X'z(s) ) b(s)] (4a)
= E,(s) a(s) + a*(s) b(s), ihg[b'(s) + (X2(S), g'2(s))b(s) + (X2(s), X'~(s))a(s)]
(4b)
= E2(s ) b(s) + A*(s) a(s),
where a prime denotes differentiation with respect to s. Note that (Xi(S), X'i(s)) is a pure imaginary quantity, and that the eigenstates Xl, X2 are defined up to an arbitrary phase (which can be chosen to be s dependent). It is easy to show that one can choose these phases to satisfy ihd ( X, (s), X'I (s)) = Ez(s ) ,
(5a)
ih~ (X2(s), X'2(s)) = E,(s).
(5b)
Indeed this choice is made below. It is convenient to rewrite eqs. (4) in matrix notation,
ihg ds \b(s)
=
a(s)-ihi(x2, x])
with E,2(s ) =- E, (s) - E2(s ).
-E,2(s)
b(s)]'
(6)
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D. Lubin et al. / Zener dynamics beyond Zener's assumptions
The matrix in eq. (6) can be considered as a "dynamical Hamiltonian", H d ( s ) , which in general differs from the "adiabatic Hamiltonian" of the problem, Ha(s ). The off-diagonal term of H d (e.g. the (1, 2) element) can be regarded as an effective coupling z~(s), which may be represented as A ( s ) = d(s) exp[i0(s)], where d and 0 are real functions of s. It is convenient to define the following variables: Y(s) ~ a(s) e i0(,)/2 ,
1)(s) =- b(s) e +i°(')/2 .
(7)
Clearly lal 2 = lal 2 and I/~[2 = ]bl 2, i.e. the probabilities to be in either state are independent of this change in phase. The resulting equations for if(s) and/~(s) read
d ihg
=
d(s, g)
-(E
z(s ) + ½hgO'(s, g))
"
(8)
Note that in the (tT,/~) representation the dynamical Hamiltonian matrix, H d, is real. In the general case the elements of H d may depend explicitly on g. As there are no approximations involved in the derivation of eq. (8) from the assumed two-level dynamics, it constitutes a general formualtion of the twolevel problem. In the remainder of this section we describe Zener's approach [1] to the solution of the dynamical problem presented by eqs. (6) or (8). The physical assumptions are the following. The crossing point is sharply defined, that is E12(s ) is large, except in the close vicinity of s = 0. The coupling term zl(s) is small compared with E~2(s ) outside the above-mentioned region. All the quantities in the problem, other than E12 , have a very slow s dependence, hence their derivatives with respect to s may be neglected. Quantitatively, this means that if we define a scale of s, denoted by Sze ., in which most of the transition process takes place, then we assume f ' ( 0 ) Sze n ~ f(0), where f is e.g. the magnitude of an off diagonal element of Hd; g is replaced by a constant (denoted by a). When s is an externally controlled parameter the latter is not necessarily an assumption but rather a specification of the problem considered. When all the above assumptions hold, H d is approximately diagonal, except in a small vicinity of s = 0. It can be assumed that the transition probability is determined by the form of H d near s = 0, while in other regions of s one may replace H 0 by another matrix which keeps the states virtually uncoupled. Specifically it is assumed that for every s we may approximate the elements of H d by the first non-vanishing term in their expansion around s = 0. Thus El2(S ) may be approximated by El2(0 ) s =-/xs, and zl(s) by a _= A(0). (The latter is valid provided that we can neglect the overlap terms (X1, X'2), (X2, X'~), as
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D. Lubin et al. / Zener dynamics beyond Zener's assumptions
well as 0'.) Having invoked all these approximations and assumptions, eq. (8) simplifies to
i h a -~s \ b(s)
(9)
b(s) / "
Eq. (9) is the model solved by Z e n e r in actuality. Since all transitions are assumed to occur in a small neighborhood of s = 07 one can find the various transition amplitudes by considering a p r e p a r e d state at s = - ~ evolving according to eq. (9) and inquire into the nature of the corresponding state at s = ~. When all of the above assumptions are justified, the answer obtained corresponds to the transition from a given state at s < 0 just outside the transition region to a state at s > 0 slightly beyond that region. As is well known (and easily seen from the matrix in eq. (9) or that in eq. (8)) the (adiabatic) eigenstates of the o p e r a t o r H are nondegenerate. For s < 0 and far enough from the transition region, the adiabatic states are basically the same as the diabatic ones; a similar statement holds for s > 0 . However, following the crossing point the diabatic state with the lower energy becomes the one with the higher energy. The higher energy adiabatic eigenstate corresponds to X2(s) (say) for s---~ - ~ and to X~(S) for s---~ + ~ and vice versa. Therefore a transition from one adiabatic state to the other means (for large [sl) staying on the same diabatic state. It is customary to define the transition amplitude between X~ and X~ as t (standing for transmission), between X2 and X2 as t', between X~ and X2 as r (reflection), and betwen X2 and X~ as r'. The transition matrix between the adiabatic levels is therefore
Clearly U is a unitary matrix, which must obey the unitary conditions Irl 2 = Ir'l 2
Itl ~ = It'l 2
Irl 2 + Itl 2 = 1
tr* = - t ' * r '
(11)
In order to obtain more detailed relations, we should go back to eq. (9) and note that it is invariant when J, b are replaced by /~*, - J * respectively, and the complex conjugate is taken. It follows that t'= t*,
r'= -r*.
(12)
The result obtained by Z e n e r for the problem formulated in eq. (9) was
It] 2 =
e-~,
A2 T -= hp~ol "
(13)
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461
The result in eq. (13) is an exact consequence of eq. (9) when initial and final conditions are specified at --- oo. In the next section we discuss how the result given by eq. (13) is modified when the conditions are different.
3. Going beyond Zener's analysis In the previous section we have set the ground for the analysis of a two-level system. We are now in a position to deviate from Z e n e r ' s original analysis, and study the ensuing modifications of the transition probabilities. Several scenarios of such modifications are discussed below. We first consider a finite integration range, and then corrections due to the s dependence of H d. 3.1.
Transition p r o b a b i l i t i e s at f i n i t e times
In this section we consider the Hamiltonian matrix of eq. (9), which is Z e n e r ' s model, but evaluate the transition probability over a finite time interval (for brevity, we call s " t i m e " in this section). The set of equations (9) was solved by Z e n e r [1]. The result in eq. (13) applies when the limits of integration are taken from s = - ~ to s = + ~. We now calculate how the result is affected when the integration is taken from s = - s 2 to s = + s I, keeping all other assumptions intact. To simplify the calculations we shall consider only finite s 1, letting - s 2 remain infinitely large. The extension to the case where s 2 is finite as well is staightforward, but the expressions obtained are m o r e complicated. There are two sources for corrections when finite times are considered. One arises from the necessity to consider the actual solution of the pertinent equations at a finite time (in this case, the finite time value of the W e b e r function, cf. ref. [1]). The second correction arises from the fact that Z e n e r ' s calculation is p e r f o r m e d in the diabatic basis. In the large Isl limit the diabatic and adiabatic states of Z e n e r ' s problem coincide. For finite times, however, the transformation from the diabatic basis to the adiabatic eigenstates should be taken into account. The first problem would be to calculate the corrections to t, r when finite times are considered. Recently, M/illen et al. [5] have estimated the time, denoted in section 2 by Sze ., required for the transition to be completed. They considered the probability of staying on the same diabatic level I (x)l 2 as a function of the dimensionless time x =-- t x s / A (Xl --= t z s l / A , Xzo, =- lZSze,/A). Starting with a value lal 2 = 1 when x is a large negative number, the evolution of lal 2 towards its x---,oo asymptotics was studied. The width of the decay region was used as the definition of Xze .. This width was obtained using the
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D. Lubin et al. / Zener dynamics beyond Zener's assumptions
scaling properties of this profile. Their results, transalated to our notation, are that Szcn = ~ (equivalently Xz~n = 1 / x / ~ ) in the sudden limit (Y ~ 1, Y was defined in eq. (13)), and SZe~ = A/k~ (Xz~" = 1) in the adiabatic limit (T~I)
#l.
Here we analyze this problem further. Following Zener, the solution of eq. (9) can be expressed in terms of Weber functions. The initial and final values of ItT(x)l2 are then determined from the asymptotic behavior of the Weber functions. In practice, the Weber function reaches is asymptotic values (corresponding to x---> -+~) to a good approximation for finite values of x, and the transition between its asymptotic values occurs in a narrow range of values of x. The value of Xz~. can be defined as the width of the region in which most of the transition takes place. Using the asymptotic analysis of the parabolic cylinder functions [7], we find that when 3' is of order unity or smaller the crossover to asymptotic behavior is when x ~ 1/x/-y, while for large -/it occurs at x = 1. We conclude that the results for the transition time obtained by both approaches agree. We now assume that indeed x 1 is large enough so that we are in the asymptotic regime, and compute the leading corrections to the transition probability. In that case we may follow Zener's original analysis, up to the last step, where we compute not only the asymptotic term, but also the leading correction to it [8]. We obtain
1~7[2 = e - ~
1+
1
1
+4x~
V ~ e-3~,/4 Re(eXp{i[27x~ + 87 l n ( x / v ~ ) + 7]/4} ) xl r * ( - i ~ , / 2 + 1) .
(14)
The general form of [ff(x)l 2 therefore consists of a sum of a slowly varying (monotonously decreasing) function, approaching the asymptotic Z e n e r value, and a fast oscillating term, whose magnitude decays to zero. Consider again the limiting cases: In the sudden limit (Y ~ 1) the amplitude of the oscillatory correction term is proportional to V~/x. At the crossover to the asymptotic regime (x = Xzcn) the magnitude of the oscillatory part of ItT[2 is comparable to 1/~[2 ~'~-'Ti"yand should not be neglected. In the adiabatic limit ( y >>1), the amplitude of the oscillations is proportional to e x p ( - ' r r 7 / 2 ) / x . In this limit, however, the term 1/4x~ dominates the results up to exponentially large times (x I >> exp('rry/2)). For intermediate values of 3' the amplitude of the oscillations at the crossover is of order unity. We conclude therefore that if y ~< 1 the ~ Defining a "superadiabatic'" basis, M.V. Berry has recently shown that the transition time in the slow rate limit is given by hX/'ff~ [6].
D. Lubin et al. / Zener dynamics beyond Zener's assumptions
463
oscillations of [~l z are significant at x = Xze.. If y is much larger than unity, the asymptotic transition probability is not yet attained at x = Xzen. The second source of corrections to Zener's formula is due to the transformation between the diabatic and adiabatic bases. The integration of eq. (9) (or the approximate solution eq. (14)) yields the (time dependent) transition coefficients (in the diabatic basis) t ~ Ill exp(i0,) and r ~ Irl exp(i0r). Earlier (cf. the analysis at the end of section 2) we have defined the transition coefficients at asymptotically large times. At finite times one has to distinguish between transition coefficients in the scheme of diabatic states, and those in the scheme of adiabatic states. We note that there is a non-trivial, time dependent, transformation between the schemes. We denote the corresponding transmission coefficient in the adiabatic representation by t a. The time dependent tansformation between the diabatic to the adiabatic bases is readily calculated by diagonalizing the Hamiltonian in eq. (9) (note that in Zener's approximation H d = Ha). In order to obtain corrections to the same order as those previously discussed, we expand the transformation matrix in powers of 1 / x 1. The result is
( 1)
Ita[2=It[ 2 1--~X~
1
Irlltl
+ 4x~-- - - x l cos(O, - Or).
(15)
(Note that the phases 0t, 0r increase with Xl, therefore the last correction term is oscillatory.) We obtain therefore a very similar result to the previous correction discussed: for ranges of values of x beyond the crossover value ( X z e n ) , ]ta]2 has a correction to its asymptotic value that decays like 1/x 2, and another oscillatory term which decays like 1 / x . The former correction is of the same order of magnitude as the one found before (cf. eq. (14)), but with an opposite sign. The amplitude of the oscillatory term has the same y dependence as the amplitude of the oscillatory term in eq. (14). These two types of corrections are independent of each other, and the complete result for the transition probability is obtained by substituting 1~1 from eq. (14) for It[ in eq.
(15). Note that in the extreme sudden limit ( y - - ~ ) , when Irl =0, there are no oscillations, but there are still corrections (of both types) of order x -2 to
Itl: l. Estimating Xzen becomes especially important when we study a system whose spectrum contains more than one narrow gap. In this case it is important to know under what conditions can the large system be approximated by a set of independent, decoupled, two-level Zener transitions. A necessary condition is obviously that the time that elapses between consecutive transitions is much larger than the time required to complete the transition. Our results indicate
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D. Lubin et al. / Zener dynamics beyond Zener's assumptions
that the latter time is not necessarily given by Xze. discussed above, but may be much longer. At x = Xzen there are significant corrections to the asymptotic result. For larger times the corrections decay like a power law of time.
3.2. The role of the dynamical Hamiltonian In this section we analyze how eq. (13) is modified when not all the assumptions leading to eq. (9) are valid. The first assumption we shall relax is that the overlap term c(s)=-(Xl(s), X'2(s)) may be neglected. Indeed, unless c(s) vanishes exactly due to some symmetry of Ho(s), it is not necessarily a small quantity for any value of s. When c(s)~O, ~l(s) in eq. (6), and consequently d(s) and O(s) in eq. (8), may depend explicitly on a. In particular, for large a, d may become large, and the transition region is increased, rendering the lowest order expansion invalid. The general problem is thus rather complicated. It may still be possible, however, to employ Zener's approach even in this case, provided the value of a is not much larger than a c =- [A(s)[/hlc(s)l. When ~ satisfies this condition, d(s) is of the same order of magnitude as A(s) (or smaller), and the transition region does not increase. In that case we may still expand H j to lowest orders around the crossing point of the effective diabatic levels, which in general will be shifted from s = 0 (see eq. (8)). To illustrate the possible consequences of this modification we consider a simple example: assume that c(s) is nearly constant, that is while we may not neglect the derivatives of Xi(s), we may do so for c(s). As a result 0 is constant, and H~ is expanded around s = 0 for every a. In that case we end up with a matrix of a similar form to the one solved by Zener, and his solution applies to this problem too. The crucial difference is that the effective coupling in the definition of y in eq. (13) is now = ,1(0) = A(0) - io~hc(0).
(16)
In this example we consider again asymptotically large times. Following Zener's analysis, the asymptotic transition probability, It[ 2, is
Itl 2 = e x p [ -
~ \(IA]2 ha
_2im(Ac,)+ha[c]2)]
is no longer monotonous in a. In of a: a = a c. In this case
fact
It] 2 has a maximal value for
It]max = exp - ~ - [IA]lc[ - Im(Ac*)]
.
(17)
afinite
value
(is)
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465
It is important to note that the value of a for which the non-monotonous behavior is obtained is in the regime of values of a for which the approximations used are valid. The condition for Itlmax 2 = 1 is C[A I = ialcl. Unlike in the original Z e n e r formula, in which It[--> 1 as a - - ~ , the corrected formula predicts Itl--~0 in this limit. In summary, the transition probability Itl 2 increases from It[ = 0 at a = 0 to 2 ]/tmax a t a c, and decreases for a > a c to Itl = 0 at a = ~ . For a >>a c the approximation breaks down. The above result is exact only for constant A and c. However, as long as the variation of A, c with s is not too rapid, changing a will shift the crossing point slightly. This will modify the effective value of y, but the same qualitative behavior will be observed. As another simple example, we consider the case where indeed c(s) =- O, but = A does depend on s. To make things even simpler, yet illustrative, we demonstate that such an s dependence may have profound consequences by considering a specific family of functions ,~(s), which is convenient for analysis, i.e. ,~(s)= d exp[i0(s)], where d is a real constant, and only the phase 0 is s-dependent. All the systems in this family have exactly the same adiabatic spectrum; therefore Zener's assumptions are of equal validity for every one of them. If a is the same in every case, the Z e n e r formula predicts exactly the same transmission probability for all members of this family. We shall see, however, that in spite of this similarity, the s dependence of 0 causes a non-universal behavior of the transition probabilities. Consider now eq. (8) with constant d and O(s). One obvious implication of eq. (8) is that a nonzero value of O'(s -- 0) produces a finite shift in the crossing point of the effective diabatic levels. For some choices of O(s) there may be more than one crossing point, or none at all. The value of the slope It depends of course on the position of the crossing. When one takes E~2(s ) = Its as an exact relation valid for every s and a constant O'(s) = [3, the crossing point is shifted from s = 0 to s = ha/3/2it, and nothing else changes. In this case the result for the transition probability is identical to that of Zener's, up to a shift of the time in which the transition occurs. If one keeps the assumption that E12(s ) = Its, but now assumes 0 ' = 2/3s, the diagonal terms in eq. (2)) read: +-(It +ha/3)s. Consequently, in this case, Z e n e r transition probability is obtained by replacing It in eq. (13) by (It + ha~3):
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(19)
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4. Discussion
In this work we have analyzed in detail the Zener problem of transitions between two adiabatic energy levels, and found corrections and deviations from Zener's analysis. We have shown that when finite times are considered there are significant corrections to the transition probability. When the s dependence of the various elements in the system is taken into account, significant corrections may arise. By considering the derivatives of the diabatic states and keeping all the other assumptions intact, we obtain a qualitative change in the behavior of the transmission probability: it is nonmonotonous in the driving rate. Finally we have shown that two-level systems with identical spectrum, driven at the same rate, can show significantly different behavior, depending on the details of the system. The conclusion here is that the validity of the Z e n e r assumptions cannot be determined merely from the structure of the adiabatic spectrum, but one has to consider the full details of the specific system at hand. One of the important applications of the two-level calculation is to the analysis of a more realistic many-level quantum system. The reduction of the N-level problem to a set of many two-level problems follows two assumptions. The first is transitions from a given level to levels not adjacent to it are neglected, and practically we have only two-level couplings. The second assumption, which is pertinent only when the first one is satisfied, is that these two level systems are uncoupled, and each is described by Zener's dynamics. In this paper we have concentrated on the second problem: the corrections and deviations from the two-level analysis, assuming that the first assumption holds. Recently we have analyzed the problem of a metallic ring threaded by a time dependent magnetic flux [4]. In that problem there is a many-level adiabatic spectrum (see fig. lb), the magnetic flux plays the role of the parameter s, and there are transitions among the adiabatic levels. The result of solving exactly the dynamical problem is that localization in energy space is obtained. We have found a f i n i t e localization length, consistent with transmission probabilities less than unity, even for infinitely large a. This result was obtained directly in the N-level system, and was in contradiction with the result obtained when the same system was approximated by a set of separable Z e n e r transitions. One possible source of this contradiction is the above discussed finite time effect. The integration limit is necessarily finite, and, as we have shown, finite size corrections decay only quadratically. Even in the extreme sudden limit, this will induce nonnegligible deviation from the value Itl = 1 at each transition. We note that in the ringe problem, when two diabatic states Xi, Xj cross as a function of s, the symmetry of the Hamiltonian results in c~-(Xi, X'i)-=0;
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468
D. Lubin et al. / Zener dynamics beyond Zener's assumptions
t h e r e f o r e we do n o t e x p e c t , n o r o b s e r v e , n o n m o n o t o n i c i t y o f the l o c a l i z a t i o n l e n g t h as a f u n c t i o n of c~. In c o n c l u s i o n , e v e n w h e n t r a n s i t i o n s in a m u l t i - l e v e l s y s t e m can b e a p p r o x i m a t e d b y c o n s e c u t i v e Z e n e r t r a n s i t i o n s , t h e l a t t e r s h o u l d b e c a l c u l a t e d with p r o p e r care: finite t i m e effects a n d the influence o f the r a t e of c h a n g e o f the e x t e r n a l bias, as well as t h e t i m e d e p e n d e n c e o f scalar p r o d u c t s o f w a v d f u n c t i o n s , s h o u l d be i n c l u d e d . In p r i n c i p l e o n e s h o u l d also c o n s i d e r c o r r e c t i o n s d u e to c o u p l i n g s a m o n g n o n a d j a c e n t levels.
Acknowledgements This w o r k was s u p p o r t e d in p a r t b y t h e U . S . - I s r a e l B i n a t i o n a l Science F u n d ( B S F ) , a n d the M i n e r v a F o u n d a t i o n , M u n i c h , F e d . R e p . G e r m a n y .
References [1] C. Zener, Proc. R. Soc. London A 137 (1932) 696; L. Landau, Soy. Phys. 1 (1932) 89; E.G.C. Stueckelberg, Helv. Phys. Acta 5 (1932) 369. [2] Y. Gefen and D.J. Thouless, Phys. Rev. Lett. 59 (1987) 1752. [3] G. Blatter and D.A. Browne, Phys. Rev. B 37 (1988) 3856. [4] D. Lubin, Y. Gefen and I. Goldhirsch, Phys. Rev. B 41 (1990) 4441. [5] K. Miillen, E. Ben-Jacob, Y. Gefen and Z. Schuss, Phys. Rev. Lett. 62 (1989) 2543. [6] M.V. Berry, Histories of Adiabatic Quantum Transitions, University of Bristol preprint (1989). [7] See e.g.: M. Abramowitz and I.A. Stegun, eds. Handbook of Mathematical Functions, p. 685. [8] See e.g.: E.T. Whittaker and G.N. Watson, A Course of Modern Analysis (Cambridge Univ. Press, Cambridge, 1952) p. 348. [9] S. Fishman, K. M/illen and E. Ben-Jacob, Phys. Rev. A, to be published.