Nonadiabatic transitions in Landau-Zener grids: integrability and semiclassical theory
NNonadiabatic transitions in Landau-Zener grids: integrability and semiclassical theory
Rajesh K. Malla, Vladimir Y. Chernyak,
2, 3 and Nikolai A. Sinitsyn Theoretical Division, and the Center for Nonlinear Studies,Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA Department of Chemistry, Wayne State University, 5101 Cass Ave, Detroit, Michigan 48202, USA Department of Mathematics, Wayne State University, 656 W. Kirby, Detroit, Michigan 48202, USA Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Dated: January 13, 2021)We demonstrate that the general model of a linearly time-dependent crossing of two energy bandsis integrable. Namely, the Hamiltonian of this model has a qaudratically time-dependent commutingoperator. We apply this property to four-state Landau-Zener (LZ) models that have previously beenused to describe the Landau-St¨uckelberg interferometry experiments with an electron shuttlingbetween two semiconductor quantum dots. The integrability then leads to simple but nontrivialexact relations for the transition probabilities. In addition, the integrability leads to a semiclassicaltheory that provides analytical approximation for the transition probabilities in these models for allparameter values. The results predict a dynamic phase transition, and show that similarly-lookingmodels belong to different topological classes.
I. INTRODUCTION
The Landau-Zener (LZ) model describes an evolu-tion for amplitudes of two states with a time-dependentHamiltonian H = (cid:18) b t gg ∗ b t (cid:19) , (1)where b , are called slopes of diabatic levels and g isthe inter-level coupling. The basis in which the off-diagonal elements of H ( t ) are time-independent is calleddiabatic basis. The LZ formula provides an exact an-alytical expression for the probability to remain in thesame diabatic state after the evolution during time t ∈ ( −∞ , + ∞ ): P LZ = e − π | g | / | b − b | . (2)This formula plays a special role in the theory of nonadi-abatic transitions because it can be used as an approxi-mation when the energy levels are mostly well separated.The adiabaticity is then broken only in disjoint regionsof time-energy, in which the nonadiabatic dynamics isexperienced only by pairs of states and the parametertime-dependence can be linearized.For nanoscale systems of modern interest, however,many states may experience the nonadiabatic transitionssimultaneously, even when the linear approximation ofthe parameter time-dependence near the nonadiabatictransitions is still applicable. The state evolution is thendescribed by the nonstationary Schr¨odinger equation i ddt | ψ (cid:105) = H ( t ) | ψ (cid:105) , (3)and the time-dependent Hamiltonian of a multistateLandau-Zener (MLZ) process has generally the form [1] H ( t ) = Bt + A, (4) FIG. 1. The linearly time-dependent diabatic levels in themodel of two crossing bands are forming a pattern that we callLZ-grid. The first and the second bands have, respectively, N and M parallel levels. The diabatic basis states of the sameband do not interact with each other directly but any such astate can be coupled directly to arbitrary diabatic states ofthe other band. where A and B are time-independent matrices, and B isdiagonal. Let E be the diagonal part of A . The nonzeroelements of Bt + E are called diabatic energies and thecorresponding eigenstates are called diabatic states. As t → ±∞ , the diabatic states coincide with the Hamil-tonian eigenstates. The goal of the MLZ theory is tofind the amplitudes S nm and the transition probabilities, P m → n = | S nm | , from the diabatic states m as t → −∞ to the states n as t → + ∞ .Among the MLZ models, there is a class of Hamilto-nians that has attracted special attention previously. Itcorresponds to the time-dependent crossing of two bandswith parallel diabatic levels, as shown in Fig. 1. Let N and M be the integer numbers of the parallel levels inthese bands. Matrices A and B then have the dimen- a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n sions ( N + M ) × ( N + M ), and A = (cid:18) E GG † E (cid:19) , B = (cid:18) b N b M (cid:19) , (5)where 1 N and 1 M are the unit, respectively, N × N andM × M matrices. The diagonal matrices E ≡ diag { e , e , . . . , e N } , E ≡ diag { e , e , . . . , e M } are responsible for the spacing between the parallel dia-batic levels; b , are the slopes of the bands. The lowerindices 1 and 2 in b i , , e i , refer, respectively, to the N -level and M -level bands. G is a N × M matrix thatdescribes direct coupling between the two bands. All el-ements of G can be nonzero and complex-valued.The band crossing MLZ model (5) was discussed orig-inally in relation to physics of the Rydberg atoms [2, 3].Optical realization of this model was used to create anoptical Galton board [4]. The early work sometimes re-ferred to the level crossing pattern in the two-band modelas to LZ-grid of energy levels. Later, LZ-grids attractedattention in relation to the two-state systems that arecoupled to an environment, which splits the two levels ofthe LZ model into the two bands of many parallel levels[5–9], and more recently LZ-grids emerged in the study ofqubits coupled to optical modes, such as in circuit QEDsystems [10–15].Although certain facts about the LZ-grids have beendetermined analytically [5], such systems remain gener-ally unsolvable. Due to the complex oscillatory behaviorof the transition probabilities as functions of the param-eters, physics of LZ-grids remains poorly studied. Ap-proximations have been developed but only for limits ofeither very small [16] or very large [17] separations of theparallel levels in the bands. Such limits are approachedvery slowly with decreasing/growing e i , , so they usu-ally give too crude approximations to realistic choices ofthe parameters. There is also one fully solvable LZ-gridmodel [18] but it does not clarify many questions aboutthe general problem.Our article has two goals. First, we add to the an-alytical understanding of the transition probabilities inall LZ-grids by applying the recent developments on thetime-dependent integrability [19, 20]. Thus, in section II,we show that all LZ-grids are formally integrable, in thesense that there is an analytic expression for the time-quadratic-polynomial operator that commutes with theHamiltonian, and satisfies an additional condition thatis needed for the integrability of time-dependent Hamil-tonians as defined in [19]. However, unlike many knownsolvable cases [21–24], LZ-grids generally remain not fullysolvable. Instead, the integrability leads to a simple butnontrivial symmetry for the transition probabilities.Our second goal is to apply this symmetry to a specificfour-state LZ-grid model that has attracted attention re-cently due to experiments with Landau-St¨uckelberg in-terferometry [25] in coupled quantum dots [26, 27]. Wewill assume that the electrostatic energy of one of the dots is changing linearly with time and, in section III,show that the integrability in such models leads to non-trivial exact relations between different transition prob-abilities.Finally, one of the applications of the integrability isthe possibility to reduce the order of the differential equa-tion for amplitudes of LZ-grid states. This property leadsto asymptotically exact expressions for leading exponentsthat describe the transition probabilities in the nearly-adiabatic limit, as it has been recently demonstrated forthe general three-state LZ model [20]. In section IV weapply this semiclassical approach to the four-state quan-tum dot models in order to derive approximate expres-sions for the experimentally most relevant probability toremain in the same quantum dot after a linear sweep ofthe gate voltage. II. INTEGRABILITY OF THE GENERALBAND-CROSSING MODEL
The integrability conditions for a time-dependentHamiltonian H ( t ) are defined as the possibility to finda parameter combination τ , and an analytical form of anontrivial operator H (cid:48) such that [19] ∂H∂τ − ∂H (cid:48) ∂t = 0 , (6)[ H, H (cid:48) ] = 0 . (7)Such conditions are known in the theory of solitons [28],and the existence of time-polynomial commuting oper-ators in several MLZ systems was originally noticed in[29]. Before we discuss the applications, let us first provethat the conditions (6) and (7) can be always satisfiedfor the LZ-grid Hamiltonian with A and B given by (5).Let us define a continuous family of operators H ( t, τ ) = B ( τ ) t + A ( τ ) , (8)where B ( τ ) and A ( τ ) are obtained from the original B and A by setting B ( τ ) ≡ Bτ, E ( τ ) ≡ τ E , G ( τ ) ≡ G √ τ , (9)and keeping E intact. Note that at τ = 1, A ( τ ) and B ( τ ) are the same as original A and B .Then, the pair of operators, H ( t, τ ) and H (cid:48) ( t, τ ), where H (cid:48) ( t, τ ) = ∂ τ B ( τ ) t ∂ τ A ( τ ) t − b − b ) τ A ( τ ) , (10)satisfy (6) and (7). Indeed, (6) is trivial to verify, whereas(7) leads to two independent conditions for the termsproportional to, separately, t and t :12 [ ∂ τ B ( τ ) , A ( τ )] + [ ∂ τ A ( τ ) , B ( τ )] = 0 , (11)[ ∂ τ A ( τ ) , A ( τ )] − b − b ) τ [ A ( τ ) , B ( τ )] = 0 , (12) FIG. 2. The true time-evolution path P (blue arrows) with τ = 1 and t ∈ ( −∞ , + ∞ ) can be deformed into the path P τ ,such that the horizontal part of P τ has τ = const (cid:54) = 1 (dashedblack arrows). These deformations do not change the evolu-tion matrix. Vertical legs of P τ have t = ± T with T → ∞ ,so they contribute only to the trivial adiabatic phases in theevolution matrix, and do not affect the transition probabili-ties. For the three-state LZ model, the path P τ can be chosenso that b → ∞ along the horizontal piece of this path. which can be verified by direct substitution of the τ -dependent matrices (8) and (9).Due to the satisfied integrability conditions, one candeform the integration path in the two-time space ( t, τ )without changing the evolution amplitudes. Namely, letus define the evolution operator U = ˆ T P exp (cid:18) − i (cid:90) P H ( t, τ ) dt + H (cid:48) ( t, τ ) dτ (cid:19) , (13)where ˆ T P is the path ordering operator along P inthe two-time space ( t, τ ). Equations (6)-(7) mean thatthe nonabelian gauge field with components A ( t, τ ) =( H, H (cid:48) ) has zero curvature, so the result of integrationin (13) does not change after the deformations of P thatkeep only the initial and final points of P intact [19], andavoid singularities of the τ -dependent Hamiltonians, asin Fig. 2.Let the physical evolution correspond to the changesof t from −∞ to + ∞ at τ = 1. Then, P starts at thepoint ( t, τ ) = ( −∞ , t and change τ from this point to another value, and only then perform t -evolution at fixed new τ . After this, we bring τ back to τ = 1 at t = + ∞ [19, 28].The τ -evolution at fixed t = − T → −∞ or t = T → + ∞ is strictly adiabatic due to the quadratic dependenceof the diagonal elements of H (cid:48) on t . Therefore, the tran-sition probabilities in the LZ-grid models that differ onlyby τ within the family (9), which is parametrized by τ ,are identical.This nontrivial invariance can now be used in combina-tion with a trivial symmetry that is common for all MLZmodels. Namely, by rescaling time in the Schr¨odingerequation (3), t → t/ √ τ , (14) FIG. 3. Varying the distances between parallel levels doesnot change the state-to-state transition probabilities if thearea enclosed by the diabatic levels (filled by blue color) isconserved. we cannot change the transition probabilities for the evo-lution in the interval t ∈ ( −∞ , ∞ ) at fixed τ . On theother hand, this rescaling corresponds to the change ofthe parameters in the original model (5): b , → b , /τ, E , → E , / √ τ , G → G/ √ τ . (15)Thus, the transition probabilities are independent ofthe variable transformations, simultaneously, (9) and(15). Combining them, we find that the transition prob-abilities in model (5) are invariant of a simple transfor-mation of the diagonal matrices: E → E √ τ , E → E / √ τ . (16)This is the most general exact result of our article. Thephysical meaning of this result is illustrated in Fig. 3. Fora model with only two levels in each band in this figure,there are only two independent level splittings∆ e ≡ e − e , ∆ e ≡ e − e , so Eq. (16) means that the transition probabilities de-pend only on the combination ∆ e ∆ e but not on theratio ∆ e / ∆ e . It is easy to verify that S = ∆ e ∆ e / ( b − b )has the physical meaning of an area enclosed by the dia-batic levels (the diamond plaquette in Fig. 3). Hence, wecan also formulate (16) as an invariance of the transitionprobabilities of the transformations of the LZ-grid thatpreserve the areas enclosed by the diabatic levels, as wellas the LZ parameters | G ij | / ( b − b ) if the slopes of thebands are also allowed to change.Unfortunately, the invariance under the transforma-tions (16) is not sufficient to solve the whole model, i.e.,to express the transition probabilities in terms of theknown special functions of the model’s parameters. Nev-ertheless, we will show that this symmetry strongly sim-plifies the analysis, and even leads to certain further ex-act relations for the transition probabilities when a modelhas additional discrete symmetries. FIG. 4. The antisymmetric model describes a single elec-tron with spin shuttling between two quantum dots with theHamiltonian (18). Dashed lines show pairs of energy levels ineach dot. Up and down red arrows mark the orthogonal spinstates. The splitting within each pair is the effect of an ex-ternal static magnetic field. The relative energies of localizedstates in different dots are controlled by time-dependent elec-tric gate voltage. The couplings g and γ describe, respectively,spin-conserving and spin-flipping electron tunnelings that arepermitted by the time-reversal symmetry of the system in theabsence of the magnetic field. III. QUANTUM DOT MODELS
In what follows, we will explore application of the sym-metry (16) to two models, which have been studied tosome extent for various reasons previously. We will referto these models as to symmetric and antisymmetric. Thesymmetric model has the Hamiltonian H ( t ) = e g g − e g gg g bt + e g g bt − e , (17)and the antisymmetric one has the Hamiltonian H ( t ) = e g − γ − e γ gg γ bt + e − γ g bt − e , (18)where all parameters are real. The symmetric model hasemerged previously in discussions of nonadiabatic behav-ior in MLZ systems in the large coupling limit [30].Physically, both models, (17) and (18), can describea single electron that jumps between discrete levels oftwo quantum dots. The linear potential ramp in one ofthe dots is then induced by applying a time-dependentelectric gate voltage difference between the dots, as in theexperiments [26, 27]. Each dot has two discrete energylevels. Most naturally, this happens when electronic spincan flip during the tunneling event due to the spin orbitcoupling, as shown in Fig. 4.In the antisymmetric case, with the Hamiltonian H ,the couplings g and γ describe then the spin preservingand the spin flipping tunneling between the two dots.The minus sign near γ guarantees the time-reversal in-variance of the model. In fact, the Hamiltonian H at e = e = 0 is the most general, up to gauge trans-formations, Hamiltonian that one can create for a time-reversed four-state system with spin [31], and the split-tings e , e (cid:54) = 0 are induced by applying the externalmagnetic field. Generally, different quantum dots havedifferent g-factors, which then means that e (cid:54) = e .Both models, H and H , have elementary discretesymmetries. For example, letΘ = − − . The Hamiltonian H has an elementary symmetry H ( t ) = − Θ H ( − t )Θ , which leads to the symmetry of the evolution operator U ( T | − T ) = T exp( − i (cid:82) H ( t ) dt ): U = Θ U † Θ , (19)from which follows that the amplitudes U and U inthe symmetric model are purely real. There are alsorelations between different scattering amplitudes, suchas, U = − U ∗ , U = − U ∗ , which lead to the rela-tions between the transition probabilities: P → = P → , P → = P → , P → = P → , and P → = P → . For areader interested in examples of discrete symmetry effectsin other MLZ models, we refer to Refs. [31, 32].Similarly, for the antisymmetric model, letΘ A = − i i i − i , then H ( t ) = − Θ A H ( − t )Θ A , from which follows that U and U are purely imagi-nary in this model, and there is the same set of relationsbetween the transition probabilities as for the symmetricmodel.Apart from this, the transition probability indepen-dence of the area conserving transformations (Fig. 3),leads to less intuitive constraints. Namely, since differ-ent quantum dots generally have different level splittings,i.e., e (cid:54) = e there is generally an asymmetry of dynamics in respect tothe initially chosen quantum dot. For example, the trivialsymmetry (19) does not predict any relation between theprobabilities P → and P → , as we illustrate in Fig. 5(a).However, the invariance of the transition probabilitiesof the transformations (16) allows us to tune e = e FIG. 5. Diabatic levels of the quantum dot models. (a)Dashed colored arrows show the transitions from level 2 tolevel 1 (violet) and from level 3 to level 4 (blue) with equaltransition probabilities in both the symmetric and the anti-symmetric models. Due to the difference, e (cid:54) = e , there is noobvious geometric symmetry between such transitions. (b)Two semiclassical paths in the diabatic level diagram thatcontribute to the transition probability from level 2 to level1 (red and green arrows). Due to the quantum interference,this probability is expected to show oscillatory dependence onthe area enclosed by the diabatic levels, S = e e /b . in both models without affecting the transition prob-abilities. Let us also add the gauge transformation H , → H , − b /
2, which does not change the prob-abilities either. At such values of the parameters, theHamiltonians (17) and (18) have an additional discretesymmetry. Namely, let θ = −
10 0 − . Then, H (cid:48) , = H e = e , − b / H (cid:48) , ( t ) = θH (cid:48) , ( t ) θ, from which we obtain additional relations on the tran-sition amplitudes, such as U = U ∗ and consequently P → = P → , e.t.c.. Let us now summarize all the rela-tions among the transition probabilities that follow fromsimultaneously the integrability and the discrete symme-tries of the models. Although the latter symmetries area bit different for H and H , they lead to the same re-lations between the transition probabilities: P → = P → , (20) P → = P → , (21) P → = P → , (22) P → = P → = P → = P → . (23)While for e = e , the relations (20)-(23) are conse-quences of trivial discrete symmetries, for e (cid:54) = e theyare generally the results of the integrability of the LZ-grid model. We also note that our analysis, and hence FIG. 6. Numerically found time-dependence of transitionprobabilities P → (solid curves) and P → (dashed curves)in the antisymmetric model for { e , e } = { , } (blue), { , } (red). The remaining parameters: b = 2, g = 2, and γ = 1. Asboth P → and P → saturate at the same values at t = ±∞ ,their intermediate time dynamics are different. relations (20)-(23) apply only to evolution from t = −∞ to t = + ∞ , whereas they do not apply to the proba-bilities at intermediate times, except for e = e , as weillustrate in Fig. 6. Hence, Eqs. (20)-(23) are our firstnontrivial prediction for experimental verification.The integrability of time-dependent Hamiltonians is atype of quantum symmetries that have not been studiedexperimentally previously. The experiments on Landau-St¨uckelberg interferometry provide an opportunity to de-tect the presence of such unusual quantum symmetriesby measuring the state-to-state transition probabilitiesin already available solid state and atomic systems. Thedeviations from the exact predictions would mean thepresence of the terms beyond the standard Hamiltonian(18), which may emerge either due to nonlinear time-dependence of the gate voltage or effects of the magneticfield on the tunneling amplitudes.In addition to the integrability conditions, there aresix elements of the transition probability matrix that areknown exactly and explicitly due to the no-go rule andthe Brundobler-Elser formula [1, 5, 33]. For the antisym-metric model they are P → = P → = 0 , (24) P → = P → = P → = P → = e − π ( g + γ ) /b , (25)and for the symmetric model we should replace γ → g .In addition to relations (20)-(23), (24), and (25), wecan only add the unitarity of the evolution constraints: (cid:88) n =1 P m → n = (cid:88) n =1 P n → m = 1 , ∀ m ∈ { , , , } . It turns out that many of the latter relations are not inde-pendent after we include the already mentioned relations.Thus, even having so many constraints, the matrix of thetransition probabilities has three unknown independentparameters that have to be calculated separately.Fortunately, the integrability leads to another simpli-fication of the model’s analysis. Namely, it was shownin [20] that the integrability enables a semiclassical ap-proach for estimation of the transition probabilities in thenearly-adiabatic limit. It was also noted in [20] that theanalytical formulas that are obtained by this approachoften provide a reasonable approximation for the numer-ical solutions at arbitrary values of the parameters.Hence, in the following sections we apply this semi-classical approach to our four-state models, H and H .For simplicity, here we will restrict ourselves only to thetransition probability P → = P → . This transitionprobability is the most physically interesting, first, be-cause it is the only one that is needed to estimate theprobability to remain in the same quantum dot after thetime-linear sweep of the gate voltage. Indeed, all theother needed for this probabilities are given by the exactexpressions (24) and (25). Moreover, a simple analysisshows that the probability P → should generally dom-inate over P → = P → in the adiabatic limit becausethe latter become nonzero after two, rather than one inthe case of P → , nonadiabatic overgap transitions.Second, the transition from level 2 to level 1 is non-trivial even in the limit of large separation of all levelcrossings because it is then influenced by quantum in-terference of different evolution trajectories, as we showin Fig. 5(b). The dependence of the probabilities of suchtransitions on the relative parameter values is understoodpoorly. Although numerical simulations of few-state sys-tems are easy, they show complex oscillatory behavior onthe parameters even in simplest and perturbative regimes[34], so it is hard to see a general pattern for the role ofdifferent parameters. Hence, by developing nonpertur-bative analytical description of P → in models (17) and(18) we will obtain a useful insight into the interferenceeffects in the nonadiabatic regime. IV. SEMICLASSICAL SOLUTION FOR P → = P → IN QUANTUM DOT MODELS
In order to find the transition probability P → formodels (17) and (18) in the adiabatic limit, we take ad-vantage of the τ -independence of the transition proba-bility and make the slopes of the tilted levels, 3 and 4,infinite by setting τ → ∞ , as it was done to study thethree-state MLZ model in [20]. The tilted levels crossboth the levels 1 and 2 then at time moments: t ± = ± e /b, where “ − ” is for the crossing of level 3 and “+” is for thecrossing of level 4.Let us assume that level 3 is initially populated andwe want to find the probability to end up on level 4. Fol-lowing [20], we can employ the fact that for high-slopecrossing the characteristic time of the nonadiabatic inter-actions is vanishing as δt ∼ / √ τ , whereas the nonadia-batic transitions between the diabatic states 3 and 4 take time that is independent of τ . Hence, we can separatelytreat the interactions of levels 3 and 4 with level 1, thenwith each other, and then with level 2. Specific details,however, depend on the models, which we will considerseparately. A. Symmetric model
First, we consider the Hamiltonian H ( t, τ ) in the limit τ → ∞ , and where τ -dependence is according to (9). Weintroduce the symmetric, | + (cid:105) , and the anti-symmetric, |−(cid:105) , combinations of the diabatic states 1 and 2: |±(cid:105) = 1 √ | (cid:105) ± | (cid:105) ) . (26)As τ → ∞ , both levels 3 and 4 couple only to the | + (cid:105) state and the nonadiabatic transitions to and from | + (cid:105) happen in the direct vicinity of time moments t − and t + . The corresponding coupling is g √ τ , and the slopedifference between the diabatic levels of | + (cid:105) and either | (cid:105) or | (cid:105) is bτ . The probabilities of fast transitions from | (cid:105) to | + (cid:105) near t − and from | + (cid:105) to | (cid:105) near t + are givenby the LZ formula for two state transitions: P → + = P + → = 1 − e − πg /b . In addition, during the time interval t ∈ ( t − , t + ) thesates | + (cid:105) and |−(cid:105) interact with each other, in particular,via the virtual transitions trough | (cid:105) and | (cid:105) , as explainedin [20]. Hence, the total transition probability from level3 to level 4 can be expressed via the product of the prob-abilities P → = P → + P ++ P + → = (cid:16) − e − πg /b (cid:17) P ++ , (27)where P ++ is the probability to remain in | + (cid:105) after thedynamics with a 2 × |±(cid:105) during the time interval t ∈ ( t − , t + ).Away from the points t = t ± , as τ → ∞ , the effectof virtual transitions to and from levels 1 and 2 can becalculated perturbatively. Up to the zeroth order in τ ,such transitions lead to an effective Hamiltonian in thesubspace of states |±(cid:105) : h v = − g b (cid:18) t − t − + 1 t − t + (cid:19) | + (cid:105)(cid:104) + | , (28)where the index v refers to the interactions appearing dueto the virtual transitions in the second order of pertur-bation series over 1 /τ / . The higher order correctionsdue to such transitions in the limit τ → ∞ vanish.The other contribution to the effective two-stateHamiltonian arrives from the splitting e of levels 1 and2, which mixes states |±(cid:105) : h e = e ( |−(cid:105)(cid:104) + | + | + (cid:105)(cid:104)−| ) . (29) P ++ can be determined by solving the effectiveSchr¨odinger equation with the Hamiltonian H eff = h v + h e . After rescaling time t → te /b , this equation in thebasis of states |±(cid:105) is given by i ddt | ψ (cid:105) = 1 b H s eff ( t ) | ψ (cid:105) , t ∈ ( − , , (30)where H s eff ( t ) = (cid:18) g tt − e e e e (cid:19) . (31)In order to find P ++ , we should find the amplitude ofevolution from | + (cid:105) as t → − | + (cid:105) as t → +1, andtake its absolute value squared. Note that the parameter b plays the role in (30) of an effective Planck constant.Thus, the integrability reduces the problem to a two-state system with time-dependent 2 × H s eff .It is possible to reformulate equation (30) by changing tothe variable x ∈ ( −∞ , + ∞ ), where t = tanh( x ), and dt/ ( t −
1) = dx . This makes it standard for applicationof the, so-called, Dykhne formula for the overgap transi-tion probability between two states [35, 36] in the adia-batic limit, in which overgap transitions are suppressedexponentially. This formula provides the correspondingslowest decaying exponent and its leading order prefac-tor. We will use this formula without switching from t to x because the direct application of the Dykhne formulato the evolution during t ∈ ( − ,
1) produces the samefinal result, and can be equally justified for the evolution(30).The difference of the eigenvalues of H s eff is found ana-lytically: ∆ E ( t ) = 2 (cid:112) ( e e ) ( t − + 4 g t (1 − t ) . (32)By equating this difference to zero we find the branchingpoints: t , = (cid:32) − r ± √ − r (cid:33) / , r ≡ ( e e ) g . (33)where the square root convention in ( . . . ) / is chosen sothat Im( t , ) >
0, and t corresponds to the “+” signin (33). Depending on whether the ratio r is bigger orsmaller than 1, we found two different types of the be-havior. Phase I : r <
1, i.e., e e < g . In this case, √ − r is real, so the branching points t , are purely imaginary.The transition probability P ++ can be estimated withthe standard Dykhne formula as P ++ = e − (2 /b )Im [ (cid:82) t ∆ E ( t ) dt ] , r < , (34) where the final integration point t is the imaginary rootin (33) that is closer to the real time axis. Phase II : r >
1, i.e., e e > g . Here, the branch-ing points have both real and imaginary parts. More-over, the imaginary parts are equal to each other, so boththe branch cuts are relevant because they correspond tosemiclassical evolution trajectories with comparable am-plitudes.The transition probability is given by a generalizedDykhne formula that sums the amplitudes of both tra-jectories and only then takes its absolute value squared: P ++ = (cid:12)(cid:12)(cid:12) e − ib (cid:82) t ∆ E ( t ) dt + iφ g + e − ib (cid:82) t ∆ E ( t ) dt (cid:12)(cid:12)(cid:12) , (35)where φ g is a geometric phase difference between the twotrajectories. It is of subdominant order O (1) in compar-ison to the integrals in (35), and for real Hamiltonianscan take only discrete values 0 or π . In appendix A, wecalculate this phase for both H and H and show thatfor the symmetric model φ g = 0.The integrals in (34), (35) cannot be simplified any-more. In this form, the probability P ++ is alreadymuch easier to calculate numerically than by solvingthe Schr¨odinger equation numerically directly. In ap-pendix B, we show that the integrals can be additionallysimplified for some choices of the parameters. Such casesare useful for developing the intuition about the magni-tude of P ++ and its dependence on the parameters.The result (35) is valid for g /b (cid:29)
1. Naturally, at g = 0 it makes an unphysical prediction: P ++ = 2. Inorder to adjust Eq. (35), we note that in our case t = − t ∗ , so, we haveRe (cid:20) b (cid:90) t ∆ E ( t ) dt (cid:21) = − Re (cid:20) b (cid:90) t ∆ E ( t ) dt (cid:21) , Im (cid:20) b (cid:90) t ∆ E ( t ) dt (cid:21) = Im (cid:20) b (cid:90) t ∆ E ( t ) dt (cid:21) , and the desired approximation that makes P ++ = 1 at g = 0 is P ++ ≈ cos (cid:104) Re (cid:16) b (cid:82) t t ∆ E ( t ) dt (cid:17)(cid:105) cosh (cid:104) Im (cid:16) b (cid:82) t t ∆ E ( t ) dt (cid:17)(cid:105) . (36)Finally, using (27), we obtain the desired approximationfor the four-state model H : P → ≈ (cid:16) − e − πg /b (cid:17) cos (cid:104) Re (cid:16) b (cid:82) t t ∆ E ( t ) dt (cid:17)(cid:105) cosh (cid:104) Im (cid:16) b (cid:82) t t ∆ E ( t ) dt (cid:17)(cid:105) . (37)In Fig. 7, the analytical predictions (34) and (36)are compared to the results obtained by solving theSchr¨odinger equation (30) numerically for several values FIG. 7. Probability P ++ calculated for the symmetric modelnumerically (plot markers), and the corresponding semiclassi-cal prediction by Eqs. (34) and (35) for r = 4 (black), r = √ r = 1 (red), r = 4 / r = 1 / of the parameter combination e e /g . The analyticalresult for the critical case, e e = g (the red curve),was taken from appendix Eq. (B2). Phases I and IIare clearly distinguishable in numerical simulations: inphase I, P ++ decays monotonously with increasing 1 /b ,whereas in phase II, P ++ oscillates as a function of 1 /b .Note also that P ++ increases with increasing couplingstrength g in phase I. For large g , the leading contributionto the sum, P → + P → , is dominated by the transitionprobability P → , and consequently P ++ . Hence, our re-sult agrees with the asymptotic behavior that was foundfor the symmetric model H in 30 in the limit e e (cid:28) g . B. Antisymmetric model with the Hamiltonian H For the Hamiltonian (18), there are two couplings, g and γ . The asymmetry between g and γ requires fromus to introduce new linear combinations of the states | (cid:105) and | (cid:105) : | A + (cid:105) = g | (cid:105) + γ | (cid:105) (cid:112) g + γ , | A − (cid:105) = − γ | (cid:105) + g | (cid:105) (cid:112) g + γ , such that | (cid:105) couples directly to | A + (cid:105) and | (cid:105) couplesto | A − (cid:105) . In the limit τ → ∞ , both level 3 and level 4cross the diabatic energies of | (cid:105) and | (cid:105) at time moments,respectively, t − and t + . Hence, the transition probability P → is P → = P → A + P + − P A − → , (38)where P + − is the probability of the transition from | A + (cid:105) as t → t − to | A − (cid:105) as t → t + , and the fast nonadiabatictransitions are given by the standard LZ formula: P → A + = P A − → = 1 − e − π ( g + γ ) /b . In order to find P + − , we follow the previous approach:we rescale time and obtain the analogous to (30) effective FIG. 8. Probability P + − for the antisymmetric model, cal-culated numerically (plot marker) for the evolution with theHamiltonian (39). Solid curves are the corresponding semi-classical predictions by Eq. (42) at e = e = 2, g = 2, and γ = 0 . γ = 1 (blue), γ = 1 . Schr¨odinger equation for evolution in the subspace | A ± (cid:105) during time interval t ∈ ( − ,
1) and the effective 2 × H A eff ( t ) = e e r − r + − r + t +1 − (cid:113) − r − /r − (cid:113) − r − /r − r − r + − r + t − , (39)where r ± ≡ ( g ± γ ) / ( e e ) . The corresponding distance between the adiabatic levelsof H A eff is∆ E A = 2 e e (1 − t ) (cid:113) ( t − + 2( t − r − + r . (40)This expression has almost the same structure asEq. (32) for the symmetric model. Similarly, the transi-tion from | A + (cid:105) to | A − (cid:105) requires a passage through theavoided crossing at t = 0, and there are two branchingpoints t and t that are obtained by setting ∆ E A = 0.However, we found two qualitative differences of theantisymmetric model from the symmetric one. First, theantisymmetric model does not have phase I, i.e., it alwaysleads to the phase with two equally important branchcuts at time points t , = (cid:18) − r − ± i (cid:113) r − r − (cid:19) / , (41)where the convention for “( . . . ) / ” is to keep only thesquare roots in the upper complex plane. Such roots in(41) satisfy the relation t = − t ∗ as in phase II of thesymmetric model. Hence, the antisymmetric model is al-ways in the phase with the oscillatory behavior. Second,we show in appendix A that the geometric phase for the Symmetric modelAntisymmetric model
FIG. 9. Transition probabilities P → are shown for the nu-merically exact simulations (plot markers) and campared tothe analytical approximations (solid lines) from Eqs. (37) and(43) for both the symmetric, H , and antisymmetric, H ,models. In all cases: e = e = 2. The coupling in thesymmetric model is g = 1 (black) and g = 3 (blue), and inthe antisymmetric model: g = 2, and γ = 0 . γ = 1(green). antisymmetric model is φ g = π . Hence, P + − ≈ (cid:12)(cid:12)(cid:12) e − ib (cid:82) t ∆ E A ( t ) dt − e − ib (cid:82) t ∆ E A ( t ) dt (cid:12)(cid:12)(cid:12) . (42)In Fig. 8, we provide the numerical check of (42), whichconfirms the analytical prediction, including the effect ofthe topological phase φ g = π .At g = 0, this formula predicts correctly that P + − = 0,and thus does not need further adjustments. However, wefound that a slightly better fit to numerical simulationsin the strongly nonadiabatic regime for the Hamiltonian H is given by P → ≈ (cid:16) − e − π ( g + γ ) /b (cid:17) sin (cid:104) b Re (cid:82) t ∆ E A ( t ) dt (cid:105) cosh (cid:104) b Im (cid:82) t ∆ E A ( t ) dt (cid:105) . (43)Formulas (37) and (43) become asymptotically exactin the adiabatic limit 1 /b → ∞ . It is instructive tolook also at how they perform in the strongly nonadi-abatic regime, for which they cannot be justified rigor-ously. In Fig. 9, we show that, although small deviationsfrom the numerically exact predictions are generally vis-ible, in a broad range of the parameters both formulasperform quite well. They still correctly predict multi-ple oscillations, including phases and amplitudes, evenin the nonadiabatic regime. Thus our semiclassical the-ory is sufficiently rigorous for, e.g., planning the futureexperiments without resorting to exhaustive numericalsimulations. V. DISCUSSION
The concept of integrability in MLZ theory originallywas introduced to unify various fully solvable models. Interestingly, there are also large classes of systems thatare formally integrable but cannot be fully solved. Thus,we showed that all LZ-grid models satisfy the integra-bility conditions for time-dependent Hamiltonians [19].As a result, the state-to-state transition probabilities inall such models are invariant of parameter rescaling (16)but this is not sufficient to determine these probabilitiesanalytically.Nevertheless, this symmetry is nontrivial and differentfrom the set of the previously known exact results thatwere derived for LZ-grids earlier [5]. In this sense, theintegrability [19] is akin in effect to a conservation lawfor a time-independent Hamiltonian.We are not aware of experimental studies of such sym-metries in explicitly time-dependent Hamiltonians, al-though these symmetries should be realizable in a broadclass of physical systems [32]. Given that the Landau-St¨uckelberg interferometry for a four-state antisymmetricmodel (18) has been already demonstrated in quantumdots [26, 27], the integrability of LZ-grids can also betested. Specifically, for the Hamiltonian (18), we predictnontrivial but simple-looking relations between the state-to-state transition probabilities (20)-(23) in the dynamicsinduced by a linear potential chirp.The experiment to observe such a nontrivial symmetryrequires a modification of the control and measurementprotocol in [26]. The standard Landau-St¨uckelberg inter-ferometry protocol induces periodic transitions throughthe region with nonadiabatic transitions. It leads to spec-tacular multi-dimensional plots for the state probabilitydependence on the potential sweeping rate and the du-ration of the periodic control. However, such measure-ments are influenced by all parameters of the scatteringmatrix for a single passage through the nonadiabatic re-gion. Hence, an analytical interpretation of such patternsfor multi-state quantum systems is usually problematicdue to the large number of relevant parameters.Here, we propose to explore the multistate Landau-Zener transition probabilities directly, for example, re-setting the state and then measuring it after each linearpotential chirp in a coupled quantum dot system [26].The advantage of such an experiment would be the directmeasurement of the characteristics, for which either exactor semiclassical analytical predictions can be established.By comparing the experimental data to the theory, onecan then explore such phenomena as the probability os-cillations due to completely nonadiabatic interference ef-fects that we exposed, as well as to confirm the exactconstraints that follow from the model’s integrablity.On the side of mathematical physics, we found fur-ther confirmations to the conjecture in [20] that thetime-dependent Hamiltonian integrability often leads to asemiclassical approach to calculate the leading contribu-tions to the overgap transition amplitudes in multistatesystems. This approach is an opportunity to explorethe driven quantum models with considerable complex-ity. Although generally approximate, it produces reason-able approximations even in the strongly nonadiabatic0regime, in which it cannot be rigorously justified.Our semiclassical analysis reveals that even the sim-plest driven models can show complex behavior, such asphase transitions, in which the adiabatic limit plays asimilar role to the thermodynamic limit in many-bodysystems. Thus, in the symmetric model, the interferingsemiclassical trajectories do not necessarily lead to oscil-lations of measurable characteristics. Rather there aretwo phases: one with oscillatory behavior and anotherone with a monotoneous decay of the transition proba-bility. We also found that models with similar structureand similar oscillatory behavior, can belong to differenttopological classes that are characterized by the topolog-ical phase φ g between the interfering semiclassical trajec-tories. The integrability allows us to observe and quan-tify such phenomena in relatively complex and stronglydriven quantum systems. Appendix A: Topological π -phase The π -phase between the amplitudes of different semi-classical trajectories in complex time has been known inliterature, see, e.g., in [37]. In practice, however, thisphase has been usually inferred numerically, and we donot know of a published rigorous analytical theory forits calculation. Therefore, here we provide such a the-ory, which exposes the topological nature of the π -phaseand provides a simple path for its calculation. Our mod-els with the Hamiltonians H s eff and H A eff are particularlysuitable for illustration because all calculations for themcan be performed analytically, and the results differ forthe different models.The geometric phase appears generally in quantum me-chanics when evolution is considered along a closed pathin the parameter space. In the context of the Dykhneformula, such a path can be found if we note that thephase difference between the amplitudes of the trajecto-ries, C − and C + in Fig. 10(a), that go through two dif-ferent branching points, is the same as the phase acquiredduring the evolution along a path C = ( C − ) − C + , thatis, the cyclic trajectory on the Riemann surface thatstarts and ends at t = 0 on the original real time axis. C follows C + along its direction marked in Fig. 10(a),and then switches to C − but follows it in the oppositedirection to what is marked in Fig. 10(a). In Fig. 10(b)we show that this path is a knot that winds around theorigin points of the branch cuts.Any real two-state time-dependent Hamiltonian can bewritten in the form H ( t ) = f ( t )1 + Z ( t ) σ z + X ( t ) σ x , (A1)where f ( t ), Z ( t ), and X ( t ) are functions of time. Let usthen definesin θ = X ( t ) (cid:112) X ( t ) + Z ( t ) , cos θ = Z ( t ) (cid:112) X ( t ) + Z ( t ) . (A2) Re(t)Im(t) (a) (c)t=0 + (b) FIG. 10. (a) The two trajectories, C − and C + , around thecorresponding branch points t and t , that make compara-ble contributions to the semiclassical probability amplitude.(b) The path C = ( C − ) − C + is topologically equivalent toa knot that winds around the branching points t , . (c) Theintegral over the knot path in Eq. (A5) is the sum of contri-butions from the infinitely small circular contours that windin opposite directions around the poles at t , . Along the cyclic path in complex plane, C , an eigen-state of H ( t ) can still be parametrized by an angle θ : | u ( θ ) (cid:105) = (cid:18) − sin θ/ θ/ (cid:19) . (A3)The topological phase originates from the fact that theHamiltonian is periodic function of θ , whereas the eigen-state vector (A3) depends on cos( θ/
2) and sin( θ/ θ changes from 0 to 2 π during an adiabaticevolution, the Hamiltonian returns to its initial form atthe end but the state vector can change sign: | u (2 π ) (cid:105) θ ( C )=2 π = −| u (0) (cid:105) . (A4)In other words, the state vector acquires a phase π .Generally, a periodic adiabatic evolution of parameterschanges the angle θ by 2 πn , where n is an arbitrary inte-ger. The topological phase is then either 0, if n is even,or π , if n is odd. There cannot be another type of ageometric phase in this case because (cid:104) u ( θ ) | ∂u ( θ ) ∂θ (cid:105) = 0.For a closed trajectory C , which can be parametrizedby time t , the topological phase is given by φ g = 12 (cid:90) C dθdt dt. (A5)Note that dθdt = 1cos θ d sin θdt . For the symmetric model (30), we have Z = 2 g t ( t − , X = e e , so,12 dθdt = r ( t + 1)4 t + r (1 − t ) = ( t + 1) r ( t − t )( t − t ) , (A6)1where r = e e /g , and t , are the two roots in the uppercomplex half-plane, which are written in (33).Note that the integrand in (A6) has simple poles at t , rather than the branching points. Hence, the integralover C is given by the difference of the residues at thesepoles, as illustrated in Fig. 10(c): φ g = πi (cid:34) Res (cid:18) dθdt (cid:19) t − Res (cid:18) dθdt (cid:19) t (cid:35) , (A7)where Res ( . . . ) a is the residue of the expression at a sim-ple pole a , and where the minus sign is because C windsaround t and t in opposite directions. Substituting theroots from (33) to (A7), and making sure that t = − t ∗ ,we find for the effective two-state system with the Hamil-tonian H s eff that φ g = 0 . (A8)Analogously, for the effective Hamiltonian (39) thatcorresponds to the antisymmetric model (18), we have dθdt = − (cid:113) r − r − t ( t − t )( t − t ) , (A9)where t , are given by (41), and t = − t ∗ . Substituting(A9) and (41) into (A7), we find that for the effectiveHamiltonian (39) the topological phase is φ g = π. (A10)Thus, despite similar oscillatory behavior, the symmetricand antisymmetric models are characterized by the dif-ferent values of topological phase φ g . In this sense themodels belong to different topological classes. Appendix B: Analytically simple special cases1. Symmetric model: the critical point at e e = g The condition e e = g marks the phase transitionpoint between phase I and phase II. In this special casethe two branching points are degenerate and purely imag-inary, i.e., t = t = i . The adiabatic energy differencesimplifies to ∆ E = 2 g t + 11 − t . Substituting t = ix into the integral in the Dykhne for-mula (34) we find2Im (cid:20)(cid:90) t ∆ E dt (cid:21) = 2 g (cid:90) − x x dx = 2( π − . Following [37], we should set the exponential prefactorin this special case to be 2 rather than 1 because thebehavior near the zero is ∆ E ∼ ( t − t ) rather than the typical ∆ E ∼ ( t − t ) / . This leads to the transitionprobability P ++ e e = g = 2 e − π − g /b . (B1)This formula is valid only asymptotically in the limit g /b (cid:29)
1. Without affecting the behavior in this do-main, we can also write P ++ e e = g ≈ π − g /b ) , (B2)which is still generally approximate but has right value,1, at g = 0. Finally, the semiclassical formula for thetransition probability in the original 4-state model forthis special case is P e e = g → ≈ (cid:16) − e − πg /b (cid:17) cosh(2( π − g /b ) . (B3)
2. Symmetric model: the case with oscillations at e e = g √ Substituting e e = g √
2, to Eq. (32), we find∆ E = 2 √ g √ t + 11 − t . The two branch points in the upper plane are then givenby t = e i π/ , t = e iπ/ . To calculate the integral, we set t = e iπ/ s, which leads to:1 b (cid:90) t dt ∆ E = 2 √ e iπ/ g b (cid:90) ds √ − s − is = g ( ξ + iζ ) b , where the analytic expressions for ξ and ζ can be writtenin terms of the known special functions but the expres-sions are a bit lengthy. We just give their numericalvalues, which can be found to very high precision: ξ = 1 . . . . , ζ = 1 . . . . . The analogous calculation leads to1 b (cid:90) t dt ∆ E = g ( − ξ + iζ ) b , and then to P ++ ≈ cos[ ξg /b ] cosh[ ζg /b ] , which provides the leading exponent for small b and is 1at g = 0.2
3. Antisymmetric model: the case with e e = g − γ In the antisymmetric model there is also a special case e e = g − γ , r − = 1 , such that the integrals can be computed explicitly. Sub-stituting r − = 1 in (40), we find the eigenvalue difference∆ E A = 2 e e (1 − t ) (cid:113) t − r , and the branch points in the upper half plane are givenby t = | r − | / e i π/ , t = | r − | / e iπ/ . In order to calculate the integral we set t = | r − | / e iπ/ s, and find1 b (cid:90) t dt ∆ E A = 2 e e | r − | / e iπ/ b (cid:90) ds √ − s − i | r − | / s = e e b { F [ | − r | ] + iG [ | − r | ] } , where analytical expressions for F [ x ] and G [ x ] can bewritten in terms of the known special functions, whichcan be calculated with high precision. Similarly, we eval-uate the integral for the branching point t :1 b (cid:90) t dt ∆ E A = e e b {− F [ | − r | ] + iG [ | − r | ] } . Combining the two contributions, and taking into ac- count the topological phase φ g = π , we finally find: P + − ≈ sin[ e e F [ | − r | ] /b ] cosh[ e e G [ | − r | ] /b ] . ACKNOWLEDGEMENTS
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