PRL 110, 150603 (2013)
week ending 12 APRIL 2013
PHYSICAL REVIEW LETTERS
Nonadiabatic Transitions in Exactly Solvable Quantum Mechanical Multichannel Model: Role of Level Curvature and Counterintuitive Behavior N. A. Sinitsyn Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Received 28 January 2013; published 10 April 2013) We derive an exact solution of an explicitly time-dependent multichannel model of quantum mechanical nonadiabatic transitions. In the limit N 1, where N is the number of states, we find that the survival probability of the initially populated state remains finite despite an almost arbitrary choice of a large number of parameters. This observation proves that quantum mechanical nonadiabatic transitions among a large number of states can effectively keep memory about the initial state of the system. This property can lead to a strongly nonergodic behavior even in the thermodynamic limit of some systems with a broad distribution of coupling constants and the lack of energy conservation. DOI: 10.1103/PhysRevLett.110.150603
PACS numbers: 05.70.Ln, 03.65.Yz, 03.75.Kk, 87.15.ht
Quantum mechanical nonadiabatic transitions have been studied in molecular collisions for a long time [1–3]. Relatively recently, multichannel nonadiabatic processes became the focus of research in condensed matter physics due to the interest in control of quantum many-body states in electronics, magnetic systems, and Bose condensates [4–7]. Condensed matter physicists often achieve insight into their systems by finding a proper model from a large variety of exactly solvable systems with a well-defined thermodynamic limit. Unfortunately, most of the known models correspond to the stationary Schro¨dinger equation. Exact solutions of explicitly time-dependent systems with a large number of states are rare. The multistate version of the Stueckelberg-MajoranaLandau-Zener (LZ) model [8,9] is one of the most investigated explicitly time-dependent problems [10–14]. Its goal is to find transition probabilities among N discrete states induced during the time evolution from ¼ 1 to ¼ þ1 in systems described by the Scho¨dinger equation with linearly time-dependent coefficients: i
dc ^ c; ¼ ðA^ þ BÞ d
(1)
where A^ and B^ are constant N N matrices. Matrix B^ is diagonal. Eigenstates of B^ are called the diabatic states and off-diagonal elements of the matrix A^ are called the coupling constants. One has to find the scattering N N ^ which element Snn0 is the amplitude of the matrix S, diabatic state n0 at ! þ1, given that at ! 1 the ^ Pnn0 ¼ system was at the state n. The related matrix P, 2 jSnn0 j , is called the matrix of transition probabilities. A number of exact results in which the regime N 1 can be effectively considered, such as the Demkov-Osherov [12] model, are currently known. Available exact results seem to indicate that transition probabilities in the model (1) behave somewhat ‘‘classically’’: All known exact solutions with a finite number of 0031-9007=13=110(15)=150603(5)
states can be either reduced to independent two-state systems [13], or can be completely understood in terms of individual pairwise LZ transitions that happen in a chronological order between each pair of diabatic states [10,12,14]. Even beyond Eq. (1), Ostrovsky studied a similar model to (1) with nonlinear level crossings and came to similar conclusions [15]. Numerical simulations show deviations from the semiclassical behavior in nonintegrable cases but the available exact results and numerical simulations of the model (1) with large sizes of the phase space [16] suggest that dynamics of such driven multistate systems can be at least qualitatively and approximately understood in terms of the successive two-state interactions that happen at moments when pairs of diabatic energies cross or pass close to each other. Pairwise transition amplitudes then can be used to estimate amplitudes of the full trajectories in the phase space. Based mostly on the available exact solutions of (1), it has been argued in various contexts [7] that if a level with an extremal slope crosses diabatic energies of a large number N of initially unpopulated states then the probability P0!0 to remain in the initial state after all intersections decreases exponentially with N, i.e., P0!0 expðNÞ, where is a coefficient, which characterizes a transition probability at a typical encountered avoided crossing point. Such a result would be perfectly justified in classical stochastic kinetics [17]. Quantum mechanical exact results only extended the validity of such semiclassical expectations. There can certainly be exceptions that correspond to resonances that can encounter when a large number of parameters are finetuned but this does not change the general qualitative understanding. In this Letter, we provide an explicit example, which unambiguously demonstrates that the above intuition about the large-N limit of quantum mechanical nonadiabatic transitions becomes incorrect if we consider the evolution beyond the linear time dependence of diabatic energies.
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Ó 2013 American Physical Society
We will show that even if N is large and all parameters in the Hamiltonian are chosen almost arbitrarily, e.g., from some random distribution, the probability to remain at the initial level after all level intersections can remain Oð1Þ, i.e., essentially finite. To achieve our goal, we will solve a model of (N þ 1) interacting states with the evolution equation for amplitudes i
N X d k2 a¼ aþ gn bn ; d n¼1
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PHYSICAL REVIEW LETTERS
PRL 110, 150603 (2013)
i
d b ¼ n bn þ gn a; d n
2it
N X db0 ¼ ðk2 iÞb0 þ gn bn ; dt n¼1
We then introduce the anzatz Z dueiut Bn ðuÞ; bn ðtÞ ¼
n ¼ 0; . . . ; N;
A
(2)
where n ¼ 1; . . . ; N. We will call the diabatic state with the amplitude a the 0th level. Its diabatic energy (i.e., the energy that it would have at zero coupling to other states) decays according to the Coulomb law, 1=, while diabatic energies of other states change linearly, as shown in Fig. 1. For this property, we will call the model (2) the LZC model, after Landau, Zener, and Coulomb. At ! 0, transitions among states do not happen because the 0th state has infinite energy. We will assume that the 0th level is populated at ! 0, while other states have vanishing amplitudes. During the evolution, for some time, distances between diabatic energies become comparable to coupling constants. This is the time when nonadiabatic transitions mostly happen. At ! þ1 all diabatic energies become again well separated and transitions terminate. From the semiclassical picture, we expect that the behavior of the transition probabilities will be mainly influenced by the encountered crossing points of the 0th level with levels n having n > 0, as shown in Fig. 1. Drawing the intuition from Ref. [7], one may also expect that the probability to remain at the initial 0th level after all intersections will be exponentially suppressed at large N. We will solve the LZC model by changing variables as a ¼ b0 and t ¼ 2 =2, leading to
(3)
db i n ¼ n bn þ gn b0 : dt
(4)
where A is a contour such that the integrand vanishes at this contour limits. Substituting (4) in (3) we obtain a 1st order differential equation for B0 ðuÞ, which is trivially solvable. Substituting the result back to (4) we find Z 2 b0 ðtÞ ¼ Q dueiut ðuÞð1=2Þþiðk =2Þ
A N Y
n¼1
bj ðtÞ ¼ Qgj
u þ n u
Z
du
ig2 =2 n
n
;
2 eiut ðuÞð1þik Þ=2
(5)
u þ j N Y u þ n ig2n =2n : u n¼1 A
Integrals in (5) have almost the same structure and the same initial conditions for amplitudes as in the model of Ostrovsky [15]. Following his steps, we consider a contour A that encloses branch cuts at u ¼ ðfn g; 0Þ from a large distance and goes to infinities at u ¼ i1 R, where R is a large real number. In this limit, we can disregard terms n in comparison with u, so that integrals in (5) simplify, e.g., Z 2 b0 ðtÞ ! Q eiut ðuÞð1=2Þþiðk =2Þ du: (6) A
In (6), the contour A can be transformed into the contour C in Fig. 2(b) by switching to the variable z ¼ iut and shrinking the contour to run around the branch cut of z. We can then use the formula Z 1 ðzÞ ¼ ðÞz1 e d (7) 2i sinðzÞ C to evaluate the integral. Returning to original variables: aðÞ ¼ b0 ð2 =2Þ and bn ðÞ ¼ bn ð2 =2Þ, we find that, at ¼ 0þ , all bn vanish, while a remains finite. If we set 2
eþ3i=4i argð1=2þik =2Þik Q¼ pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 1 þ ek
2
lnð2Þ=2
;
(8)
then we will arrange at ! 0þ that 2
aðÞ!0 ik ; FIG. 1 (color online). Time dependence of the eigenspectrum (adiabatic energies) of the Hamiltonian with elements H00 ¼ k2 =, Hnn ¼ n , Hn0 ¼ H0n ¼ gn , and zero otherwise (n ¼ 1; . . . ; N; N ¼ 8).
bj ðÞ!0 OðÞ;
j 0; (9)
which corresponds to the initially populated 0th level asymptotic. In order to find transition probabilities at ! þ1 limit, we continuously deform the contour A
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into a combination of contours n that inclose the branch cuts at u ¼ n as shown in Fig. 2(a). In the limit t ! þ1, only the vicinity of the branching points contribute essentially to each integral over n . Hence one can change variables u ! u þ n , keeping the dependence on u only for terms that are singular near the origin of the n th cut. In all other factors, we can substitute u by its value at this point. Simple dimensionality arguments show that the integration for bj ðtÞt!þ1 in (5) over i remains finite only if i ¼ j. Hence a jth integral in (5) over j provides the asymptotic at t ! þ1 for bj ðtÞ, i.e., Y N 2 ðj Þigj =2j b0 ðtÞ!þ1 Q j¼1
P ð1=2Þþiðk2 =2Þi N ig2 =2j j¼1 j dueiut ðuÞ ; 0 PN 2 2 bj ðtÞ!þ1 Qgj ðj Þð1=2Þþiðk =2Þi n¼1 gn =2n (10)
Z
N Y
2
ðn j Þiðgn =2n Þ
n¼1;nj
Z
0
2
dueiut ðuÞð1Þþiðgj =2j Þ ; j 0;
where we should assume that ðiÞ ¼ ei=2 and 1 ¼ ei . Remaining integrals again can be evaluated with Eq. (7). In the Supplemental Material [18] we provide explicit expressions for asymptotic amplitudes at ! þ1 that we obtained from (10). Transition probabilities can be obtained by taking squares of the absolute values of transition amplitudes. Surprisingly, the final result appears very simple looking. In order to write it, it is convenient to introduce LZ-like probabilities: 2
pj ¼ egj =jj j ;
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PHYSICAL REVIEW LETTERS
PRL 110, 150603 (2013)
j ¼ 1; . . . ; N:
(11)
The transition probabilities from the initially populated 0th level to all possible states are then given by
FIG. 2. (a) The integration contour A encloses all branch cuts (dashed lines) from a large distance. The integral over this contour can then be expressed as the sum of the integrals over contours n , n ¼ 0; . . . ; N along the branch cuts. (b) By a change of variables, each integral over n can be transformed into the integral over the contour C.
P0!0 ¼
j > 0: P0!j ¼
j < 0: P0!j ¼
Q 2 Q 0 pn Þ þ ek ð m pm Þ 1 þ ek ð
Qn >j n
pn Þð1 pj Þ
1 þ ek ð
Qn 0 pn means the product over all pn with indexes n of states having slopes n larger than zero. If there are no such states, this product is set to unity. Several qualitative observations follow straightforwardly: Corollary 1: The transition probability from the 0th level to the level j with j > 0 depends only on coupling constants and slopes of the levels n with n j . Corollary 2: The transition probability from the 0th level to the level j with j < 0 depends only on coupling constants and slopes of the levels n with n j . Corollary 1 would be expected from the semiclassical point of view. In the independent crossing approximation, only crossing points that precede the crossing of the level j would influence the population of the jth level. This situation has previously been encountered in the solution of the Demkov-Osherov model [12]. Parameter k characterizes the curvature of the 0th level near the avoided crossings [18]. The limit of large values of k corresponds to locally linear dependence of 0th diabatic energy on time, so that the formulas (12) and (13) naturally reproduce the results of the Demkov-Osherov formula [12] if we set k ! þ1. In contrast, Corollary 2 is very counterintuitive. Surprisingly, none of the encountered avoided crossing points influences transition probabilities to the levels j with j < 0, no matter how strong couplings at crossing points are. Moreover, only couplings to the levels n with n < j , i.e., the levels that diverge faster than the level j from the 0th level, contribute to the transition probability to the jth diabatic state. In Fig. 3, we compare our theoretical predictions (12)–(14) with transition probabilities obtained in our numerical studies of the evolution (3) in a three-state LZC model. We find excellent agreement between numerical and theoretical results. In particular, Fig. 3(a) shows that if 1 > 2 > 0 then the transition probability to the level with the slope 1 does not depend on the coupling g2 to the level with a lower slope, in agreement with Corollary 1. In the case when 1 > 2 , and simultaneously 2 < 0, we find that already the transition probability to the level with the slope 2 does not depend on the coupling g1 between the 0th level and the level with the slope 1 . Corollary 3: The probability to stay at the 0th level at ! þ1 remains finite if either j > 0 or j < 0 for all j ¼ 1; . . . ; N. Moreover, in the limit N ! 1 and at finite
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FIG. 3 (color online). Probabilities of diabatic states at time t ! þ1 in (2 þ 1)-LZC model for (a) 1 > 2 > 0; k2 ¼ 0:7, g1 ¼ 0:3, 1 ¼ 1, 2 ¼ 0:5; (b) 1 > 0, 2 < 0; k2 ¼ 0:5, g2 ¼ 1:0, 1 ¼ 1, 2 ¼ 1; (c) 1 < 0, 2 < 0; k2 ¼ 0:2, g2 ¼ 0:3, 1 ¼ 0:5, 2 ¼ 1. Solid curves are theoretical predictions by Eqs. (12)–(14). Discrete points represent the results of the numerical calculations. Insets on the right illustrate time dependence of diabatic energies. In all numerical tests, the time evolves from ¼ 0:0001 to ¼ 800.
couplings, the probability to remain in the 0th level depends only on the parameter k. The last observation is the main result of this Letter. It is normally expected that if a 0th level interacts with a large number of states with arbitrary coupling strengths, such an interaction would lead to a sort of equilibration. It has been often argued that the survival probability should be, actually, vanishing exponentially with the number of levels N that interact with the highest slope level [7]. The 0th
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level satisfies this condition if conditions of Corollary 3 are satisfied. Nevertheless, Eq. (12) predicts that if all j > 0 2 2 then P0!0 > ek =ð1 þ ek Þ, and if all j < 0 then 2 P0!0 > 1=ð1 þ ek Þ. In conclusion, we determined transition probabilities in a multichannel model of nonadiabatic transitions with a well-defined transition probability matrix among all diabatic states. In our model, one of the diabatic energy levels changed nonlinearly, which allowed us to investigate the effect of the curvature of this level on the transition probability matrix. We found that the behavior of the probability to remain at the initially populated level, as well as the levels having a negative slope of diabatic energy, can behave very counterintuitively. For example, some of the transition probabilities can be finite but independent of the couplings at all avoided crossing points encountered during the system’s evolution. We predict that systems with nonadiabatic transitions in the large N limit can show strongly nonclassical behavior, e.g., with some states keeping a finite amplitude despite a large distribution of coupling constants. In the Supplemental Material [18] we demonstrate numerically that this property appears also in a wide class of diabatic potentials, e.g., of the form 1=tr with some constant parameter r. The independent crossing approximation works for them only when P0!0 remains close to unity, and the exponentially strong enhancement of the probability to stay on the initial level is observed otherwise. The Coulomb potential frequently appears in studies of nonadiabatic transitions in molecular physics [1,15,19]. Moreover, if one of the levels is degenerate before a molecular collision or an application of a time-dependent field, this degeneracy experiences fanlike spitting, as in Rydberg’s atoms due to the Stark effect [20]. Hence, the LZC model represents an important class of processes in atomic and molecular physics. Among other applications, we mention that, in the limit k ! 0, the LZC model corresponds to the previously unstudied case of an N-state bow-tie model [14] with the time evolution from ¼ 0 to ¼ þ1. In the Supplemental Material [18], we show that the two-state version of the LZC model provides, at a little cost for complexity, a better approximation of the transition probability at an avoided crossing than the StueckelbergMajorana-Landau-Zener formula [21]. It also provides a sample of a solvable model of an avoided diabatic crossing with a constant off-diagonal coupling. Finally, we provide an example in Ref. [18] of a possible realization of a three-state LZC model in an interacting qubit system. This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396.
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[1] N. Rosen and C. Zener, Phys. Rev. 40, 502 (1932); J. B. Delos and W. R. Thorson, Phys. Rev. A 6, 728 (1972); 17, 1343 (1978). [2] E. E. Nikitin, Adv. Quantum Chem. 5, 135 (1970). [3] V. I. Osherov and H. Nakamura, J. Chem. Phys. 105, 2770 (1996). [4] Y. Kayanuma and K. Saito, Phys. Rev. A 77, 010101(R) (2008); S. Longhi and G. Della Valle, Phys. Rev. A 86, 043633 (2012). [5] V. A. Yurovsky, A. Ben-Reuven, and P. S. Julienne, Phys. Rev. A 65, 043607 (2002); D. Sun, A. Abanov, and V. L. Pokrovsky, Europhys. Lett. 83, 16 003 (2008); M. A. Kayali and N. A. Sinitsyn, Phys. Rev. A 67, 045603 (2003); W. H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett. 95, 105701 (2005); B. Damski and W. H. Zurek, Phys. Rev. Lett. 104, 160404 (2010); Phys. Rev. A 73, 063405 (2006); A. del Campo, M. M. Rams, and W. H. Zurek, Phys. Rev. Lett. 109, 115703 (2012); B. Damski, H. T. Quan, and W. H. Zurek, Phys. Rev. A 83, 062104 (2011); B. Damski and W. H. Zurek, Phys. Rev. Lett. 99, 130402 (2007); V. Gurarie, Phys. Rev. A 80, 023626 (2009); A. Altland, V. Gurarie, T. Kriecherbauer, and A. Polkovnikov, Phys. Rev. A 79, 042703 (2009); A. Altland and V. Gurarie, Phys. Rev. Lett. 100, 063602 (2008); A. P. Itin and P. To¨rma¨, Phys. Rev. A 79, 055602 (2009). [6] W. Wernsdorfer, R. Sessoli, A. Caneschi, D. Gatteschi, A. Cornia, and D. Mailly, J. Appl. Phys. 87, 5481 (2000); W. Wernsdorfer, A. Caneschi, R. Sessoli, D. Gatteschi, A. Cornia, V. Villar, and C. Paulsen, Phys. Rev. Lett. 84, 2965 (2000); W. Wernsdorfer, R. Sessoli, A. Caneschi, D. Gatteschi, and A. Cornia, Europhys. Lett. 50, 552 (2000). [7] K. Saito, M. Wubs, S. Kohler, Y. Kayanuma, and P. Ha¨nggi, Phys. Rev. B 75, 214308 (2007); P. Ao and J. Rammer, Phys. Rev. B 43, 5397 (1991); M. H. S. Amin, D. V. Averin, and J. A. Nesteroff, Phys. Rev. A 79, 022107 (2009); V. N. Ostrovsky and M. V. Volkov, Phys. Rev. B 73, 060405 (2006); J. Keeling, A. V. Shytov, and L. S. Levitov, Phys. Rev. Lett. 101, 196404 (2008); M. Wubs, K. Saito, S. Kohler, P. Ha¨nggi, and Y. Kayanuma, Phys. Rev. Lett. 97, 200404 (2006); M. Schecter and A. Kamenev, Phys. Rev. A 85, 043623 (2012); B. E. Dobrescu and V. L. Pokrovsky, Phys. Lett. A 350, 154 (2006); N. A. Sinitsyn, Phys. Rev. A 87, 032701 (2013). [8] E. Majorana, Nuovo Cimento 9, 43 (1932). [9] L. D. Landau, Phys. Z. Sowjetunion 2, 46 (1932); C. Zener, Proc. R. Soc. A 137, 696 (1932); E. C. G. Stu¨ckelberg, Helv. Phys. Acta 5, 369 (1932).
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[10] S. Brundobler and V. Elser, J. Phys. A 26, 1211 (1993); N. A. Sinitsyn, J. Phys. A 37, 10 691 (2004); B. E. Dobrescu and N. A. Sinitsyn, J. Phys. B 39, 1253 (2006); M. V. Volkov and V. N. Ostrovsky, J. Phys. B 37, 4069 (2004); 38, 907 (2005). [11] V. L. Pokrovsky and N. A. Sinitsyn, Phys. Rev. B 65, 153105 (2002); M. V. Volkov and V. N. Ostrovsky, Phys. Rev. A 75, 022105 (2007); A. V. Shytov, Phys. Rev. A 70, 052708 (2004); M. V. Volkov and V. N. Ostrovsky, J. Phys. B 37, 4069 (2004); A. A. Rangelov, J. Piilo, and N. V. Vitanov, Phys. Rev. A 72, 053404 (2005). [12] Yu. N. Demkov and V. I. Osherov, Zh. Eksp. Teor. Fiz. 53, 1589 (1967) [Sov. Phys. JETP 26, 916 (1968)]. [13] N. A. Sinitsyn, Phys. Rev. B 66, 205303 (2002); J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005). [14] Yu. N. Demkov and V. N. Ostrovsky, Phys. Rev. A 61, 032705 (2000); J. Phys. B 28, 403 (1995); V. N. Ostrovsky and H. Nakamura, J. Phys. A 30, 6939 (1997); Yu. N. Demkov and V. N. Ostrovsky, J. Phys. B 34, 2419 (2001); C. E. Carroll and F. T. Hioe, J. Phys. A 19, 1151 (1986). [15] V. N. Ostrovsky, Phys. Rev. A 68, 012710 (2003). [16] N. A. Sinitsyn and V. V. Dobrovitski, Phys. Rev. B 70, 174449 (2004). [17] N. A. Sinitsyn and N. Prokof’ev, Phys. Rev. B 67, 134403 (2003). [18] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.110.150603 for explicit expressions for asymptotic amplitudes at ! þ1 obtained from (10). [19] R. S. Tantawi, A. S. Sabbah, J. H. Macek, and S. Y. Ovchinnikov, Phys. Rev. A 62, 042710 (2000); J. S. Cohen, L. A. Collins, and N. F. Lane, Phys. Rev. A 17, 1343 (1978); T. R. Dinterman and J. B. Delos, Phys. Rev. A 15, 463 (1977). [20] J. S. Cabral, J. M. Kondo, L. F. Gonc¸alves, L. G. Marcassa, D. Booth, J. Tallant, and J. P. Shaffer, New J. Phys. 12, 093023 (2010); F. Baumgartner and H. Helm, Phys. Rev. Lett. 104, 103002 (2010); J. M. Menendez, I. Martin, and A. M. Velasco, J. Chem. Phys. 119, 12 926 (2003); V. A. Nascimento, L. L. Caliri, A. Schwettmann, J. P. Shaffer, and L. G. Marcassa, Phys. Rev. Lett. 102, 213201 (2009); F. Robicheaux, C. Wesdorp, and L. D. Noordam, Phys. Rev. A 62, 043404 (2000); Y.-L. He, J. Phys. B 45, 015001 (2012). [21] H. Nakamura, Nonadiabatic Transition (World Scientific, Singapore, 2012), 2nd ed.
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PHYSICAL REVIEW LETTERS
Nonadiabatic Transitions in Exactly Solvable Quantum Mechanical Multichannel Model: Role of Level Curvature and Counterintuitive Behavior N. A. Sinitsyn Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Received 28 January 2013; published 10 April 2013) We derive an exact solution of an explicitly time-dependent multichannel model of quantum mechanical nonadiabatic transitions. In the limit N 1, where N is the number of states, we find that the survival probability of the initially populated state remains finite despite an almost arbitrary choice of a large number of parameters. This observation proves that quantum mechanical nonadiabatic transitions among a large number of states can effectively keep memory about the initial state of the system. This property can lead to a strongly nonergodic behavior even in the thermodynamic limit of some systems with a broad distribution of coupling constants and the lack of energy conservation. DOI: 10.1103/PhysRevLett.110.150603
PACS numbers: 05.70.Ln, 03.65.Yz, 03.75.Kk, 87.15.ht
Quantum mechanical nonadiabatic transitions have been studied in molecular collisions for a long time [1–3]. Relatively recently, multichannel nonadiabatic processes became the focus of research in condensed matter physics due to the interest in control of quantum many-body states in electronics, magnetic systems, and Bose condensates [4–7]. Condensed matter physicists often achieve insight into their systems by finding a proper model from a large variety of exactly solvable systems with a well-defined thermodynamic limit. Unfortunately, most of the known models correspond to the stationary Schro¨dinger equation. Exact solutions of explicitly time-dependent systems with a large number of states are rare. The multistate version of the Stueckelberg-MajoranaLandau-Zener (LZ) model [8,9] is one of the most investigated explicitly time-dependent problems [10–14]. Its goal is to find transition probabilities among N discrete states induced during the time evolution from ¼ 1 to ¼ þ1 in systems described by the Scho¨dinger equation with linearly time-dependent coefficients: i
dc ^ c; ¼ ðA^ þ BÞ d
(1)
where A^ and B^ are constant N N matrices. Matrix B^ is diagonal. Eigenstates of B^ are called the diabatic states and off-diagonal elements of the matrix A^ are called the coupling constants. One has to find the scattering N N ^ which element Snn0 is the amplitude of the matrix S, diabatic state n0 at ! þ1, given that at ! 1 the ^ Pnn0 ¼ system was at the state n. The related matrix P, 2 jSnn0 j , is called the matrix of transition probabilities. A number of exact results in which the regime N 1 can be effectively considered, such as the Demkov-Osherov [12] model, are currently known. Available exact results seem to indicate that transition probabilities in the model (1) behave somewhat ‘‘classically’’: All known exact solutions with a finite number of 0031-9007=13=110(15)=150603(5)
states can be either reduced to independent two-state systems [13], or can be completely understood in terms of individual pairwise LZ transitions that happen in a chronological order between each pair of diabatic states [10,12,14]. Even beyond Eq. (1), Ostrovsky studied a similar model to (1) with nonlinear level crossings and came to similar conclusions [15]. Numerical simulations show deviations from the semiclassical behavior in nonintegrable cases but the available exact results and numerical simulations of the model (1) with large sizes of the phase space [16] suggest that dynamics of such driven multistate systems can be at least qualitatively and approximately understood in terms of the successive two-state interactions that happen at moments when pairs of diabatic energies cross or pass close to each other. Pairwise transition amplitudes then can be used to estimate amplitudes of the full trajectories in the phase space. Based mostly on the available exact solutions of (1), it has been argued in various contexts [7] that if a level with an extremal slope crosses diabatic energies of a large number N of initially unpopulated states then the probability P0!0 to remain in the initial state after all intersections decreases exponentially with N, i.e., P0!0 expðNÞ, where is a coefficient, which characterizes a transition probability at a typical encountered avoided crossing point. Such a result would be perfectly justified in classical stochastic kinetics [17]. Quantum mechanical exact results only extended the validity of such semiclassical expectations. There can certainly be exceptions that correspond to resonances that can encounter when a large number of parameters are finetuned but this does not change the general qualitative understanding. In this Letter, we provide an explicit example, which unambiguously demonstrates that the above intuition about the large-N limit of quantum mechanical nonadiabatic transitions becomes incorrect if we consider the evolution beyond the linear time dependence of diabatic energies.
150603-1
Ó 2013 American Physical Society
We will show that even if N is large and all parameters in the Hamiltonian are chosen almost arbitrarily, e.g., from some random distribution, the probability to remain at the initial level after all level intersections can remain Oð1Þ, i.e., essentially finite. To achieve our goal, we will solve a model of (N þ 1) interacting states with the evolution equation for amplitudes i
N X d k2 a¼ aþ gn bn ; d n¼1
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PHYSICAL REVIEW LETTERS
PRL 110, 150603 (2013)
i
d b ¼ n bn þ gn a; d n
2it
N X db0 ¼ ðk2 iÞb0 þ gn bn ; dt n¼1
We then introduce the anzatz Z dueiut Bn ðuÞ; bn ðtÞ ¼
n ¼ 0; . . . ; N;
A
(2)
where n ¼ 1; . . . ; N. We will call the diabatic state with the amplitude a the 0th level. Its diabatic energy (i.e., the energy that it would have at zero coupling to other states) decays according to the Coulomb law, 1=, while diabatic energies of other states change linearly, as shown in Fig. 1. For this property, we will call the model (2) the LZC model, after Landau, Zener, and Coulomb. At ! 0, transitions among states do not happen because the 0th state has infinite energy. We will assume that the 0th level is populated at ! 0, while other states have vanishing amplitudes. During the evolution, for some time, distances between diabatic energies become comparable to coupling constants. This is the time when nonadiabatic transitions mostly happen. At ! þ1 all diabatic energies become again well separated and transitions terminate. From the semiclassical picture, we expect that the behavior of the transition probabilities will be mainly influenced by the encountered crossing points of the 0th level with levels n having n > 0, as shown in Fig. 1. Drawing the intuition from Ref. [7], one may also expect that the probability to remain at the initial 0th level after all intersections will be exponentially suppressed at large N. We will solve the LZC model by changing variables as a ¼ b0 and t ¼ 2 =2, leading to
(3)
db i n ¼ n bn þ gn b0 : dt
(4)
where A is a contour such that the integrand vanishes at this contour limits. Substituting (4) in (3) we obtain a 1st order differential equation for B0 ðuÞ, which is trivially solvable. Substituting the result back to (4) we find Z 2 b0 ðtÞ ¼ Q dueiut ðuÞð1=2Þþiðk =2Þ
A N Y
n¼1
bj ðtÞ ¼ Qgj
u þ n u
Z
du
ig2 =2 n
n
;
2 eiut ðuÞð1þik Þ=2
(5)
u þ j N Y u þ n ig2n =2n : u n¼1 A
Integrals in (5) have almost the same structure and the same initial conditions for amplitudes as in the model of Ostrovsky [15]. Following his steps, we consider a contour A that encloses branch cuts at u ¼ ðfn g; 0Þ from a large distance and goes to infinities at u ¼ i1 R, where R is a large real number. In this limit, we can disregard terms n in comparison with u, so that integrals in (5) simplify, e.g., Z 2 b0 ðtÞ ! Q eiut ðuÞð1=2Þþiðk =2Þ du: (6) A
In (6), the contour A can be transformed into the contour C in Fig. 2(b) by switching to the variable z ¼ iut and shrinking the contour to run around the branch cut of z. We can then use the formula Z 1 ðzÞ ¼ ðÞz1 e d (7) 2i sinðzÞ C to evaluate the integral. Returning to original variables: aðÞ ¼ b0 ð2 =2Þ and bn ðÞ ¼ bn ð2 =2Þ, we find that, at ¼ 0þ , all bn vanish, while a remains finite. If we set 2
eþ3i=4i argð1=2þik =2Þik Q¼ pffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 4 1 þ ek
2
lnð2Þ=2
;
(8)
then we will arrange at ! 0þ that 2
aðÞ!0 ik ; FIG. 1 (color online). Time dependence of the eigenspectrum (adiabatic energies) of the Hamiltonian with elements H00 ¼ k2 =, Hnn ¼ n , Hn0 ¼ H0n ¼ gn , and zero otherwise (n ¼ 1; . . . ; N; N ¼ 8).
bj ðÞ!0 OðÞ;
j 0; (9)
which corresponds to the initially populated 0th level asymptotic. In order to find transition probabilities at ! þ1 limit, we continuously deform the contour A
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into a combination of contours n that inclose the branch cuts at u ¼ n as shown in Fig. 2(a). In the limit t ! þ1, only the vicinity of the branching points contribute essentially to each integral over n . Hence one can change variables u ! u þ n , keeping the dependence on u only for terms that are singular near the origin of the n th cut. In all other factors, we can substitute u by its value at this point. Simple dimensionality arguments show that the integration for bj ðtÞt!þ1 in (5) over i remains finite only if i ¼ j. Hence a jth integral in (5) over j provides the asymptotic at t ! þ1 for bj ðtÞ, i.e., Y N 2 ðj Þigj =2j b0 ðtÞ!þ1 Q j¼1
P ð1=2Þþiðk2 =2Þi N ig2 =2j j¼1 j dueiut ðuÞ ; 0 PN 2 2 bj ðtÞ!þ1 Qgj ðj Þð1=2Þþiðk =2Þi n¼1 gn =2n (10)
Z
N Y
2
ðn j Þiðgn =2n Þ
n¼1;nj
Z
0
2
dueiut ðuÞð1Þþiðgj =2j Þ ; j 0;
where we should assume that ðiÞ ¼ ei=2 and 1 ¼ ei . Remaining integrals again can be evaluated with Eq. (7). In the Supplemental Material [18] we provide explicit expressions for asymptotic amplitudes at ! þ1 that we obtained from (10). Transition probabilities can be obtained by taking squares of the absolute values of transition amplitudes. Surprisingly, the final result appears very simple looking. In order to write it, it is convenient to introduce LZ-like probabilities: 2
pj ¼ egj =jj j ;
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j ¼ 1; . . . ; N:
(11)
The transition probabilities from the initially populated 0th level to all possible states are then given by
FIG. 2. (a) The integration contour A encloses all branch cuts (dashed lines) from a large distance. The integral over this contour can then be expressed as the sum of the integrals over contours n , n ¼ 0; . . . ; N along the branch cuts. (b) By a change of variables, each integral over n can be transformed into the integral over the contour C.
P0!0 ¼
j > 0: P0!j ¼
j < 0: P0!j ¼
Q 2 Q 0 pn Þ þ ek ð m pm Þ 1 þ ek ð
Qn >j n
pn Þð1 pj Þ
1 þ ek ð
Qn 0 pn means the product over all pn with indexes n of states having slopes n larger than zero. If there are no such states, this product is set to unity. Several qualitative observations follow straightforwardly: Corollary 1: The transition probability from the 0th level to the level j with j > 0 depends only on coupling constants and slopes of the levels n with n j . Corollary 2: The transition probability from the 0th level to the level j with j < 0 depends only on coupling constants and slopes of the levels n with n j . Corollary 1 would be expected from the semiclassical point of view. In the independent crossing approximation, only crossing points that precede the crossing of the level j would influence the population of the jth level. This situation has previously been encountered in the solution of the Demkov-Osherov model [12]. Parameter k characterizes the curvature of the 0th level near the avoided crossings [18]. The limit of large values of k corresponds to locally linear dependence of 0th diabatic energy on time, so that the formulas (12) and (13) naturally reproduce the results of the Demkov-Osherov formula [12] if we set k ! þ1. In contrast, Corollary 2 is very counterintuitive. Surprisingly, none of the encountered avoided crossing points influences transition probabilities to the levels j with j < 0, no matter how strong couplings at crossing points are. Moreover, only couplings to the levels n with n < j , i.e., the levels that diverge faster than the level j from the 0th level, contribute to the transition probability to the jth diabatic state. In Fig. 3, we compare our theoretical predictions (12)–(14) with transition probabilities obtained in our numerical studies of the evolution (3) in a three-state LZC model. We find excellent agreement between numerical and theoretical results. In particular, Fig. 3(a) shows that if 1 > 2 > 0 then the transition probability to the level with the slope 1 does not depend on the coupling g2 to the level with a lower slope, in agreement with Corollary 1. In the case when 1 > 2 , and simultaneously 2 < 0, we find that already the transition probability to the level with the slope 2 does not depend on the coupling g1 between the 0th level and the level with the slope 1 . Corollary 3: The probability to stay at the 0th level at ! þ1 remains finite if either j > 0 or j < 0 for all j ¼ 1; . . . ; N. Moreover, in the limit N ! 1 and at finite
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FIG. 3 (color online). Probabilities of diabatic states at time t ! þ1 in (2 þ 1)-LZC model for (a) 1 > 2 > 0; k2 ¼ 0:7, g1 ¼ 0:3, 1 ¼ 1, 2 ¼ 0:5; (b) 1 > 0, 2 < 0; k2 ¼ 0:5, g2 ¼ 1:0, 1 ¼ 1, 2 ¼ 1; (c) 1 < 0, 2 < 0; k2 ¼ 0:2, g2 ¼ 0:3, 1 ¼ 0:5, 2 ¼ 1. Solid curves are theoretical predictions by Eqs. (12)–(14). Discrete points represent the results of the numerical calculations. Insets on the right illustrate time dependence of diabatic energies. In all numerical tests, the time evolves from ¼ 0:0001 to ¼ 800.
couplings, the probability to remain in the 0th level depends only on the parameter k. The last observation is the main result of this Letter. It is normally expected that if a 0th level interacts with a large number of states with arbitrary coupling strengths, such an interaction would lead to a sort of equilibration. It has been often argued that the survival probability should be, actually, vanishing exponentially with the number of levels N that interact with the highest slope level [7]. The 0th
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level satisfies this condition if conditions of Corollary 3 are satisfied. Nevertheless, Eq. (12) predicts that if all j > 0 2 2 then P0!0 > ek =ð1 þ ek Þ, and if all j < 0 then 2 P0!0 > 1=ð1 þ ek Þ. In conclusion, we determined transition probabilities in a multichannel model of nonadiabatic transitions with a well-defined transition probability matrix among all diabatic states. In our model, one of the diabatic energy levels changed nonlinearly, which allowed us to investigate the effect of the curvature of this level on the transition probability matrix. We found that the behavior of the probability to remain at the initially populated level, as well as the levels having a negative slope of diabatic energy, can behave very counterintuitively. For example, some of the transition probabilities can be finite but independent of the couplings at all avoided crossing points encountered during the system’s evolution. We predict that systems with nonadiabatic transitions in the large N limit can show strongly nonclassical behavior, e.g., with some states keeping a finite amplitude despite a large distribution of coupling constants. In the Supplemental Material [18] we demonstrate numerically that this property appears also in a wide class of diabatic potentials, e.g., of the form 1=tr with some constant parameter r. The independent crossing approximation works for them only when P0!0 remains close to unity, and the exponentially strong enhancement of the probability to stay on the initial level is observed otherwise. The Coulomb potential frequently appears in studies of nonadiabatic transitions in molecular physics [1,15,19]. Moreover, if one of the levels is degenerate before a molecular collision or an application of a time-dependent field, this degeneracy experiences fanlike spitting, as in Rydberg’s atoms due to the Stark effect [20]. Hence, the LZC model represents an important class of processes in atomic and molecular physics. Among other applications, we mention that, in the limit k ! 0, the LZC model corresponds to the previously unstudied case of an N-state bow-tie model [14] with the time evolution from ¼ 0 to ¼ þ1. In the Supplemental Material [18], we show that the two-state version of the LZC model provides, at a little cost for complexity, a better approximation of the transition probability at an avoided crossing than the StueckelbergMajorana-Landau-Zener formula [21]. It also provides a sample of a solvable model of an avoided diabatic crossing with a constant off-diagonal coupling. Finally, we provide an example in Ref. [18] of a possible realization of a three-state LZC model in an interacting qubit system. This work was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE-AC52-06NA25396.
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[1] N. Rosen and C. Zener, Phys. Rev. 40, 502 (1932); J. B. Delos and W. R. Thorson, Phys. Rev. A 6, 728 (1972); 17, 1343 (1978). [2] E. E. Nikitin, Adv. Quantum Chem. 5, 135 (1970). [3] V. I. Osherov and H. Nakamura, J. Chem. Phys. 105, 2770 (1996). [4] Y. Kayanuma and K. Saito, Phys. Rev. A 77, 010101(R) (2008); S. Longhi and G. Della Valle, Phys. Rev. A 86, 043633 (2012). [5] V. A. Yurovsky, A. Ben-Reuven, and P. S. Julienne, Phys. Rev. A 65, 043607 (2002); D. Sun, A. Abanov, and V. L. Pokrovsky, Europhys. Lett. 83, 16 003 (2008); M. A. Kayali and N. A. Sinitsyn, Phys. Rev. A 67, 045603 (2003); W. H. Zurek, U. Dorner, and P. Zoller, Phys. Rev. Lett. 95, 105701 (2005); B. Damski and W. H. Zurek, Phys. Rev. Lett. 104, 160404 (2010); Phys. Rev. A 73, 063405 (2006); A. del Campo, M. M. Rams, and W. H. Zurek, Phys. Rev. Lett. 109, 115703 (2012); B. Damski, H. T. Quan, and W. H. Zurek, Phys. Rev. A 83, 062104 (2011); B. Damski and W. H. Zurek, Phys. Rev. Lett. 99, 130402 (2007); V. Gurarie, Phys. Rev. A 80, 023626 (2009); A. Altland, V. Gurarie, T. Kriecherbauer, and A. Polkovnikov, Phys. Rev. A 79, 042703 (2009); A. Altland and V. Gurarie, Phys. Rev. Lett. 100, 063602 (2008); A. P. Itin and P. To¨rma¨, Phys. Rev. A 79, 055602 (2009). [6] W. Wernsdorfer, R. Sessoli, A. Caneschi, D. Gatteschi, A. Cornia, and D. Mailly, J. Appl. Phys. 87, 5481 (2000); W. Wernsdorfer, A. Caneschi, R. Sessoli, D. Gatteschi, A. Cornia, V. Villar, and C. Paulsen, Phys. Rev. Lett. 84, 2965 (2000); W. Wernsdorfer, R. Sessoli, A. Caneschi, D. Gatteschi, and A. Cornia, Europhys. Lett. 50, 552 (2000). [7] K. Saito, M. Wubs, S. Kohler, Y. Kayanuma, and P. Ha¨nggi, Phys. Rev. B 75, 214308 (2007); P. Ao and J. Rammer, Phys. Rev. B 43, 5397 (1991); M. H. S. Amin, D. V. Averin, and J. A. Nesteroff, Phys. Rev. A 79, 022107 (2009); V. N. Ostrovsky and M. V. Volkov, Phys. Rev. B 73, 060405 (2006); J. Keeling, A. V. Shytov, and L. S. Levitov, Phys. Rev. Lett. 101, 196404 (2008); M. Wubs, K. Saito, S. Kohler, P. Ha¨nggi, and Y. Kayanuma, Phys. Rev. Lett. 97, 200404 (2006); M. Schecter and A. Kamenev, Phys. Rev. A 85, 043623 (2012); B. E. Dobrescu and V. L. Pokrovsky, Phys. Lett. A 350, 154 (2006); N. A. Sinitsyn, Phys. Rev. A 87, 032701 (2013). [8] E. Majorana, Nuovo Cimento 9, 43 (1932). [9] L. D. Landau, Phys. Z. Sowjetunion 2, 46 (1932); C. Zener, Proc. R. Soc. A 137, 696 (1932); E. C. G. Stu¨ckelberg, Helv. Phys. Acta 5, 369 (1932).
week ending 12 APRIL 2013
[10] S. Brundobler and V. Elser, J. Phys. A 26, 1211 (1993); N. A. Sinitsyn, J. Phys. A 37, 10 691 (2004); B. E. Dobrescu and N. A. Sinitsyn, J. Phys. B 39, 1253 (2006); M. V. Volkov and V. N. Ostrovsky, J. Phys. B 37, 4069 (2004); 38, 907 (2005). [11] V. L. Pokrovsky and N. A. Sinitsyn, Phys. Rev. B 65, 153105 (2002); M. V. Volkov and V. N. Ostrovsky, Phys. Rev. A 75, 022105 (2007); A. V. Shytov, Phys. Rev. A 70, 052708 (2004); M. V. Volkov and V. N. Ostrovsky, J. Phys. B 37, 4069 (2004); A. A. Rangelov, J. Piilo, and N. V. Vitanov, Phys. Rev. A 72, 053404 (2005). [12] Yu. N. Demkov and V. I. Osherov, Zh. Eksp. Teor. Fiz. 53, 1589 (1967) [Sov. Phys. JETP 26, 916 (1968)]. [13] N. A. Sinitsyn, Phys. Rev. B 66, 205303 (2002); J. Dziarmaga, Phys. Rev. Lett. 95, 245701 (2005). [14] Yu. N. Demkov and V. N. Ostrovsky, Phys. Rev. A 61, 032705 (2000); J. Phys. B 28, 403 (1995); V. N. Ostrovsky and H. Nakamura, J. Phys. A 30, 6939 (1997); Yu. N. Demkov and V. N. Ostrovsky, J. Phys. B 34, 2419 (2001); C. E. Carroll and F. T. Hioe, J. Phys. A 19, 1151 (1986). [15] V. N. Ostrovsky, Phys. Rev. A 68, 012710 (2003). [16] N. A. Sinitsyn and V. V. Dobrovitski, Phys. Rev. B 70, 174449 (2004). [17] N. A. Sinitsyn and N. Prokof’ev, Phys. Rev. B 67, 134403 (2003). [18] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.110.150603 for explicit expressions for asymptotic amplitudes at ! þ1 obtained from (10). [19] R. S. Tantawi, A. S. Sabbah, J. H. Macek, and S. Y. Ovchinnikov, Phys. Rev. A 62, 042710 (2000); J. S. Cohen, L. A. Collins, and N. F. Lane, Phys. Rev. A 17, 1343 (1978); T. R. Dinterman and J. B. Delos, Phys. Rev. A 15, 463 (1977). [20] J. S. Cabral, J. M. Kondo, L. F. Gonc¸alves, L. G. Marcassa, D. Booth, J. Tallant, and J. P. Shaffer, New J. Phys. 12, 093023 (2010); F. Baumgartner and H. Helm, Phys. Rev. Lett. 104, 103002 (2010); J. M. Menendez, I. Martin, and A. M. Velasco, J. Chem. Phys. 119, 12 926 (2003); V. A. Nascimento, L. L. Caliri, A. Schwettmann, J. P. Shaffer, and L. G. Marcassa, Phys. Rev. Lett. 102, 213201 (2009); F. Robicheaux, C. Wesdorp, and L. D. Noordam, Phys. Rev. A 62, 043404 (2000); Y.-L. He, J. Phys. B 45, 015001 (2012). [21] H. Nakamura, Nonadiabatic Transition (World Scientific, Singapore, 2012), 2nd ed.
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