SEMI-CLASSICAL MODELS FOR THE SCHRÖDINGER EQUATION WITH PERIODIC POTENTIALS AND BAND CROSSINGS. Lihui Chai. Shi Jin. and Qin Li

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1 Manuscript submitted to AIMS Journals Volume X, Number X, XX X doi:.3934/... pp. X XX SEMI-CLASSICAL MODELS FO THE SCHÖDINGE EQUATION WITH PEIODIC POTENTIALS AND BAND COSSINGS Lihui Chai Department of Mathematical Science Tsinghua University Bejing 84, China Shi Jin Department of Mathematics, Institute of Natural Sciences, and MOE Key Lab in Scientific and Engineering Computing Shanghai Jiao Tong University Shanghai 4, China and Department of Mathematics University of Wisconsin-Madison Madison, WI 5376, USA and Qin Li Department of Mathematics University of Wisconsin-Madison Madison, WI 5376, USA (Communicated by the associate editor name Abstract. The Bloch decomposition plays a fundamental role in the study of quantum mechanics and wave propagation in periodic media. Most of the homogenization theory developed for the study of high frequency or semi-classical limit for these problems assumes no crossing of the Bloch bands, resulting in classical Liouville equations in the limit along each Bloch band. In this article, we derive semi-classical models for the equation in periodic media that take into account band crossing, which is important to describe quantum transitions between Bloch bands. Our idea is still based on the Wigner transform (on the Bloch eigenfunctions, but in taking the semi-classical approimation, we retain the off-diagonal entries of the Wigner matri, which cannot be ignored near the point of band crossing. This results in coupled inhomogeneous Liouville systems that can suitably describe quantum tunneling between bands that are not well-separated. We also develop a domain decomposition method that couples these semi-classical models with the classical Liouville equations (valid away from zones of band crossing for a multiscale computation. Solutions of these models are numerically compared with those of the Schröding equation to justify the validity of these new models for band-crossings. Mathematics Subject Classification. Primary: 58F5, 58F7; Secondary: 53C35. Key words and phrases. equation, Periodic, Semi-classical models, Inter-band transition. This work was partially supported by NSF grants DMS-4546 and NSF DMS NMS grant DMS-79.

2 L. CHAI, S. JIN AND Q. LI. Introduction. The linear equation with a periodic potential is an important model in solid state physics. It describes the motion of electrons in a crystal with a lattice structure. We consider the following one dimensional equation ( ( iεφ ε t = ε φε + V + U( φ ε, t >,. ( ε Here, φ ε is the comple-valued wave function, ε is the dimensionless rescaled Planck constant, V ( is a periodic potential with the lattice L = πz, so V ( + ν = V ( for any and ν L. U( is a smooth eternal potential function. We denote the dual lattice of L by L and L = Z. The fundamental domain of L is (, π and the first Brillouin zone is B = (,. In the semi-classical regime ε, the equation without a periodic potential has a semi-classical limit governed by the Liouville equation [35]. For the equation with a periodic potential, the Liouville equation can also be obtained along each Bloch band [], and it was justified rigorously in [3, 4, 5] for the case U. The result can be generalized to the case when a weak random potential [] and nonlinear self-consistent potential [3, 4] are presented. It has been shown that in the crystal, the electrons remain in a certain quantum subsystem, move along the m-th band and the dynamics is given by ẋ = k E m (k, where E m is the energy corresponding to the m-th Bloch band [5]. Higher order correction relevant to Berry phase can be included, see [8, 9, ]. All of these results use the adiabatic assumption, namely, different Bloch bands are well-separated and there is no band-crossing. The non-adiabatic or diabatic effect should be considered whenever the transitions between energy levels of the quantum system play an important role. This may happen when the gap between the energy levels becomes small enough in comparison to the scaled Planck constant ε. The well-known Landau-Zener formula [, 37] describes the asymptotic effect of avoided crossings in various specific situations. The study of such quantum tunnelings is important in many applications, from quantum dynamics in chemical reaction [34], semiconductors to Bose-Einstein condensation [6]. While in the case of band separation there have been significant mathematical progress in understanding the semi-classical limit [, 3, 5, 8, 9], as well as numerical methods that utilizes the Bloch decomposition [7], there has been little mathematical and computational works for the band-miing case. In the contet of surface hopping method, associated with the Born-Oppenheimer approimation, the Landau-Zener phenomenon has been studied computationally by Tully etc. [34, 33, 3, ] and mathematically [5,, 3, ]. For a Liouville equation based computational approach, see [9], and a quantum-classical model for surface hopping, see [6, 6]. In this paper, we use the Wigner-Bloch theory to derive semi-classical models for the equation with periodic potentials ( that account for bandcrossing. Without band-crossing, the classical Liouville equation can be obtained along each Bloch band, as ε. This only includes the diagonal entries of the Wigner-Bloch matri (the Wigner transform of the Bloch functions which is valid away from the crossing zone. In our semi-classical models we include the leading order of the off-diagonal entries, resulting in system of comple, inhomogeneous Liouville equations. These systems contain terms that describe for transitions between bands, as well as Berry phase information which is related to the quantum Hall effect [3]. These systems are still hyperbolic, with oscillatory forcing terms

3 SEMI-CLASSICAL MODELS FO BAND COSSINGS 3 that converge (in the weak sense to zero in the semi-classical limit away from band-crossing zones so the classical Liouville equation can be recovered. Several numerical eperiments using these models produce numerical solutions that adequately describe the quantum transitions between bands, when compared with the solutions of the original equation (. The computational cost of these semi-classical models, while considerably lower than that of the original equation, is higher than the adiabatic Liouville equations without band-crossing. In order to further reduce the computational cost, a hybrid method that couples the classical Liouville equation away from the crossing zone to the new semi-classical models to be used in the crossing zones is introduced. A couple of other more efficient computational approaches for linear periodic potentials, are also introduced in section 5. This paper is organized as follows. In Section, we introduce the Wigner transform and the Bloch eigenvalue problem that are the two tools to study the semiclassical limit of the equation with periodic potentials. We also review the classical limit without band-crossing which gives rise to the classical Liouville equation for each Bloch band. In Section 3 we derive the new semi-classical Liouville systems that account for quantum transitions between different bands. In Section 4 we introduce a domain-decomposition based hybrid model that couples the classical Liouville system away from the band crossing zone to the new semiclassical models used in the crossing zones. Some numerical eamples are presented in Section 5 to validate these semi-classical models for quantum transitions between Bloch bands. More efficient numerical methods were introduced for simpler linear potentials. Section 6 concludes this paper.. The classical limit without band-crossing... The Wigner transform. Define the asymmetric Wigner transformation as in [], dy W ε (t,, k = π eiky φ ε (t, εy φ ε (t,. ( where φ ε is the comple conjugate of φ ε, and φ ε is the solution to the equation (, then one obtains the Wigner equation: W ε t + k W ε + iε W ε = e iµ/ε ˆV (µ [Wε (, k µ W ε (, k] iε µ L + dω iε π eiω Û(ω [W ε (, k εω W ε (, k], where Û(ω is the Fourier transform of U(y: Û(ω = dy e iωy U(y, ω, and ˆV (µ is the discrete Fourier coefficients of V (y: ˆV (µ = π π dy e iµy V (y, µ L. Here (, π is the fundamental domain of the lattice L. (3

4 4 L. CHAI, S. JIN AND Q. LI Denote z = ε as the fast variable. To separate the dependence on both the slow and the fast variables, one can write W ε (t,, k as W ε (t,, z, k, and replace the spatial derivative by + ε z. Then (3 becomes: ( t W ε + k + W ε + iε ( ε z + W ε ε z = e iµ/ε ˆV (µ [Wε (, k µ W ε (, k] (4 iε µ L + dω iε π eiω Û(ω [W ε (, k εω W ε (, k]. Assume U( is smooth enough, one throws away high order terms, and (4 becomes: W ε t + k W ε U W ε k + W ε i z = ε LW ε, (5 where the skew symmetric operator L is given by Lf(z, k = k f z + i f z i µ L e iµ/ε ˆV (µ [f(, k µ f(, k]. Asymptotically one epands W ε by: W ε (t,, k = W ( t,, ε, k + εw ( t,, ε, k + and plugs this ansatz into (4. In the order of O( ε and O( respectively, one gets: W t + k W LW =, U W k + W i z = LW. (6a (6b This implies that W is in the kernel of L. We seek a good basis of kerl in section., and leave the eact formulation of W to section.3... The Bloch eigenvalue problem. The eigenfunctions of L are constructed by studying the following eigenvalue problem: Ψ(z, p + V (zψ(z, p z = E(pΨ(z, p, (7a Ψ(z + ν, p = e ipν Ψ(z, p, ν L, (7b Ψ (z + ν, p z = Ψ eipν (z, p, ν L. z (7c With each specific p, one constructs a boundary condition (7b and (7c and solves the eigenvalue problem. Denote E m (p as the m-th eigenvalue with multiplicity r m, and Ψ α m, with α =,..., r m, as the associated α-th eigenfunction. Ψ m is usually called the m-th Bloch eigenfunction [5]. For this problem, one can easily check the following properties: (a The eigenvalues E m (p are L periodic in p and have constant finite multiplicity outside a closed zero-measure subset F of p. Outside F, one orders the eigenvalues as E (p < E (p <... < E m (p <... with E m (p as m, uniformly in p.

5 SEMI-CLASSICAL MODELS FO BAND COSSINGS 5 (b For any p (,, {Ψ α m (, p} forms a complete orthonormal basis in L (, π, i.e. (Ψ α m, Ψ β n := π dz π Ψα m(z, p Ψ β n(z, p = δ mn δ αβ. (8 (c For all φ L (, one has the following Bloch decomposition: r m φ( = c α m(pψ α m(, p dp (9 Define m α= where c α m is the Bloch coefficient: c α m(p = φ( Ψ α m(, p d. B Φ α m(z, p = e ipz Ψ α m(z, p, then Φ α m is a z periodic function with period π. For any p (,, {Φ α m (, p} forms a complete orthonormal basis in L (, π. For simplicity, throughout this paper we make the following assumption: Assumption. The multiplicity for each eigenvalue E m is outside F. We will drop the superscript α when there is no ambiguity..3. The classical limit without band-crossing. In this section, we review the classical limit without band-crossing []. For that, we need the following assumption: Assumption. The set F is a empty set, i.e. eigenvalues are strictly apart from each other everywhere in p, namely E (p < E (p <... < E j (p <.... By taking the Wigner transformation on the Bloch eigenfunctions, one obtains a basis on the phase space. Define the z periodic functions Q mn (z, k by: Q mn (z, k = Q mn (z, µ k, p k = π where k is an arbitrary real number and is decomposed as: k = p k + µ k, p k B, µ k L. Lemma.. Define the inner product, : f, g := µ L π dy π eiky Ψ m (z y, p k Ψ n (z, p k, ( dz π f(z, µḡ(z, µ, f, g L( (, π, l (L, then for any p ( /, /, {Q mn (,, p} forms a complete orthonormal basis in L ( (, π, l (L. Proof. The orthonormal condition Q mn, Q jl = δ mj δ nl, ( can be proved by simply using (8. To prove the completeness, it is sufficient to show that: if there eists an f L ( (, π, l (L, such that f, Q mn = for all m, n N, then f(z, µ. Assume f, Q mn = for all m, n N. By the definition of Q mn, µ π π dydz (π f(z, µeiµy Φ m (z y, p Φ n (z, p =, m, n N.

6 6 L. CHAI, S. JIN AND Q. LI Since {Φ n (, p} forms a complete orthonormal basis in L (, π, the above equality implies that thus µ π which implies that f(z, µ. A straightforward computation gives: dy π f(z, µeiµy Φ m (z y, p =, m N, f(z, µe iµy, µ LQ mn (z, k = LQ mn (z, µ, p k = i(e m (p E n (pq mn (z, µ, p k. ( Apparently, under Assumptions and, ( gives: Therefore, from (6a W has the form kerl = span{q mm, m =,, }. (3 W (t,, z, k = m σ mm (t,, p k Q mm (z, µ k, p k, (4 with σ mm representing the epansion coefficients. To derive the equation for them, one plugs it back into (6b, and takes the inner product with Q mm on both sides, the right hand side vanishes due to the skew symmetry of L and (: The left hand side, on the other hand, gives: LW, Q mm = W, LQ mm =. (5 t W + k W U k W + i z W, Q mm = [ ] t σ nn Q nn, Q mm + σ nn kq nn, Q mm n [ U ( p σ nn Q nn, Q mm + σ m p Q nn, Q mm ] (6 n = t σ mm + p E m σ mm U p σ mm. Thus one can combine (5 and (6, and obtain: t σ mm + p E m σ mm U p σ mm =. (7 In the derivation of (6, the following equalities were used : kq mn, Q jl = i ( δnl ( z Ψ m, Ψ j + δ mj ( z Ψ l, Ψ n, (8a p Q mn, Q jl = δ nl ( p Φ m, Φ j + δ mj (Φ l, p Φ n, (8b z Q mn, Q jl = δ nl ( z Ψ m, Ψ j δ mj ( z Ψ l, Ψ n, (8c p E m δ mj + ( p Φ m, Φ j (E m E j = i ( z Ψ m, Ψ j, (8d ( p Φ m, Φ m + (Φ m, p Φ m =. (8e The details can also be found in []. Notice that (7 is the classical Liouville equation for each Bloch band. emark. Similar results were rigorously proved in [5, 3].

7 SEMI-CLASSICAL MODELS FO BAND COSSINGS 7 3. Asymptotic models for the band-to-band transition. This section is for the case when Assumption is not satisfied. Physically, ε is small but nonzero, and electrons can tunnel across bands for all p. But this tunneling is negligible when bands are far away from each other, the so-called adiabatic assumption. This is the basis for the asymptotically epansion (4, which throws away the dependence on the band-transition terms Q mn (m n of W. However, if there eists m n and a point p c, such that E m (p c E n (p c O(ε γ with γ being a real number larger than zero, then LQ mn O(ε γ, so the asymptotic epansion (6 does not hold any more. When this happens, physically one observes significant tunneling effect. We seek asymptotic models to handle the band-to-band transition in this section. 3.. The derivation of a two-band semi-classical Liouville system using the asymmetric Wigner transform. In order to handle the band-to-band transition phenomena, we come back to the asymptotic model (5 and use the following epression (compare with (4!: W ε = m σ mm Q mm + m n σ mn Q mn. (9 Without loss of generality, we tackle a two-band problem. Define the crossing point as p c = arg min p { E E } and assume and p c =. Plugging (9 into (5 and taking the inner product with Q mn as mentioned in Sec..3, one gets a system for σ mn s: σ t = U σ + E p [( Φ p, Φ + ( Ψ i z, Ψ ( σ + Φ, Φ p σ ] σ, U σ p σ t = U σ + E p [( Φ p, Φ + ( Ψ i z, Ψ ( σ + Φ, Φ p σ ] σ, U σ p σ t = U σ + E p [( Φ, Φ p + ( Ψ i z, Ψ σ + ( Φ p, Φ σ U σ + σ p + ie E ε σ + ( Φ, Φ p σ ( Φ p, Φ ] σ, σ t = U σ + E p [( Φ, Φ p + ( Ψ i z, Ψ σ + ( Φ p, Φ σ U σ + σ p + ie E ε σ + ( Φ, Φ p σ ( Φ p, Φ ] σ.

8 8 L. CHAI, S. JIN AND Q. LI This system can be written in vector form as: t σ + A σ + B p σ = BCσ + id ε σ (a where σ = ( σ σ σ σ T (b B = U I, D = diag (, E E, E E, p E ψ A = ψ p E p E ψ, ψ p E C = φ φ φ φ φ φ φ φ φ φ φ φ, (c (d (e ψ mn = i ( z Ψ m, Ψ n, and φ mn = ( p Φ m, Φ n, (f Where the superscript T denotes the matri transpose, I is the identity matri. Generally, ψ = ψ, φ = φ are comple-valued quantities, and φ, φ are purely imaginary, so A = A is Hermitian, and C = C is anti-hermitian (where the superscript denotes the matri conjugate transpose. Note that Uφ mm is the so-called Berry-connection [7, 9, 3]. emark. According to the adiabatic theory [36, 7], the state initially in one eigenstate Φ m (p( will remain in an instantaneous eigenstate Φ m (p(t when time t is sufficiently slow. One can write the state at time t as ( φ m (t = e iγm(t ep i ε t dt E m (p(t Φ m (p(t, m =,, ( where p(t is the trajectory satisfying ṗ(t = U. The second eponential in ( is known as the dynamical phase factor, and γ m in the first eponential is the path integral of the Berry connection, i.e. γ m = i φ mm (p dp, ( which is called the Berry phase. Obviously this term cancel out in the density function φ m φm, however if considering the inter-band density φ m φn with m n, then one obtains { ( φ m (p(t φ n (p(t = ep i γ m (t γ n (t ε Φ m (p(t Φ n (p(t, t dt ( E m (p(t E n (p(t } (3

9 SEMI-CLASSICAL MODELS FO BAND COSSINGS 9 and the phase in the above equation i d ( γ m (t γ n (t t dt ( E m (p(t E n (p(t dt ε = dp ( φmm (p(t φ nn (p(t i Em (p(t E n (p(t dt ε( = U (φ mm φ nn i ( ε (E m E n = BC + id. ε In comparison with (, one can see that it is just the coefficient of the σ mn term on the right hand side of (a for σ mn. This shows that the Berry phase plays an important role in the inter-band transitions. It can be shown the system ( is a hyperbolic system (see the details in Appendi A. Since the source matri C = C, all of its eigenvalues are purely imaginary. Thus this system has unique bounded solution if the initial data are bounded [9] uniformly in ε. For later convenience, we call the semi-classical Liouville system ( system, and this is the system that will be discussed and numerically solved in the paper. One also needs to equip it with appropriate initial condition. Choose the initial data of the equation as two wave packets along the two Bloch bands in the following form [7, 7]: mn (4 ( ( φ I = a ( Φ ε, S ( e is(/ε + a ( Φ ε, S ( e is(/ε. (5 Then, the initial data of the Wigner function, for ε, has the approimation: W I (, z, k a ( W (z, k + a ( W (z, k + a (a ( ( W (z, k + W (z, k, (6 with W mn (z, k = dy π eiky Φ m (z y, S ( εy Φ n (z, S (e i(s( εy S(/ε. Using Taylor epansion on S ( εy S ( and Φ m (z y, S ( εy, one gets dy W mn (z, k = π ei(k S(y Φ m (z y, S ( Φ n (z, S ( + O(ε, then by ignoring the O(ε term, and using the periodicity of Φ m (z, p on z, one can change the integral into a summation of integrals from to π: W mn (z, k = ν L π dy π ei(k S((y+ν Φ m (z y, S ( Φ n (z, S (. Applying the equality e ikν = ν L µ L δ(k + µ,

10 L. CHAI, S. JIN AND Q. LI one gets W mn (z, k = µ L δ(k + µ S = µ L δ(p k + µ S = µ L δ(p k + µ S π π π π = δ(p k + µ S µ L = ( pk + µ S Qmn (z, µ k, p k. µ L δ dy π ei(k Sy Φ m (z y, S Φ n (z, S dy π eiky Ψ m (z y, S Ψ n (z, S dy π eiky Ψ m (z y, p k + µ Ψ n (z, p k + µ dy π eiky Ψ m (z y, p k Ψ n (z, p k In the above derivation, from the second line to the third line, we use the fact that δ(p p f(p g(p dp = f(p g(p = δ(p p f(pg(p dp to replace the argument S to p k + µ. Without loss of generality, we assume that S ( /, /, then (7 becomes W mn (z, k = δ ( p k S Qmn (z, µ k, p k. (8 Compare (8 with (6, one has the initial data for σ: (7 σ(,, p = δ (p S ( ( a a a a a a T. (9 3.. The semi-classical Liouville system using the symmetric Wigner transform. All the analysis above is done in the framework of the asymmetric Wigner transform (. One could also use the symmetric Wigner transform: Wε s (t,, k = dy π eiky φ ε ( t, εy φε ( t, + εy The derivation is similar, thus we skip the details, and give a list of the results:. The Wigner equation corresponding (3 is Wε s + k W ε s t = [ e iµ/ε ˆV (µ Wε s iε µ L + dω iε π eiω Û(ω (, k µ. ( Wε s, k + µ ] [ ( Wε s, k εω ( Wε s, k + εω ].. Corresponding to the asymptotic Wigner equation for asymmetrical transformation (5, one has: Wε s + k W ε s t U Wε s k = ε Ls Wε s, (3 where the skew symmetric operator L s is given by L s f(z, k = k f z e iµ/ε ˆV (µ [f(, k µ/ f(, k + µ/]. i µ L

11 SEMI-CLASSICAL MODELS FO BAND COSSINGS 3. Corresponding to (6, one has the O ( ε and O( epansions: W s t + k W s L s W s =, (3a U W s k = Ls W s. (3b 4. Same as in (, one has the following symmetrical definition for Q mm Q s mn(z, k = Q s mn(z, µ, p = π They are eigenfunctions of L s dy ( π ei(p+µy Ψ m z y (, p Ψn z + y, p. L s Q s mn = i(e m E n Q s mn. 5. If the eigenvalues {E n } are well separated, i.e. E m E n, for m n, the solution to (3a is: W s = m σ s mmq s mm. By taking the inner product with Q s mm on both sides of (3b, one obtains the same classical Liouville equations for σ s mm as in (7 t σ s mm + p E m σ s mm U p σ s mm =. 6. If some bands touch at point p c, the solution to (3 is given by: W s ε = m σ mm Q s mm + m n σ mn Q s mn. In the two-band case, m, n =,, then σ s mn is governed by, t σ s + A s σ s + B s p σ s = B s C s σ s + ids ε σs, (3 where σ s = ( σ s σ s σ s σ s T, then B s = B, C s = C and D s = D are the same as the asymmetric case, while A s is given by p E ψ ψ A s = ψ E +E p ψ ψ E +E p ψ ψ ψ p E and ψ mn, φ mn are given by (f. Noted that A s = (A s. We call this new system obtained by the symmetric Wigner transform (3 the Liouville-S system. Apparently the only difference from lies in the transport matrices A A s. But they share the same weak limit as ε (see Appendi B for detail. This formally suggests the similar behavior of σ, σ and σ, s σ, s which is confirmed by numerical results in Section A multiscale domain decomposition method.

12 L. CHAI, S. JIN AND Q. LI 4.. Asymptotic behavior of the two-band semi-classical Liouville systems. Away from p c, E and E are well-separated, and the i ε terms for the transition coefficients σ and σ in /S lead to high oscillations, thus as ε, the system formally goes to its weak limit: the classical one (7. For ε >, around p c, however, both σ and σ become significant, and the band-to-band transition is no longer negligible. In fact, based on the distance to the crossing point p c, one could obtain some asymptotic properties of the transition coefficients σ mn (m n. Assume that the initial data for the transition coefficients are all zero, and that U does not change sign in time for all, take U > for eample, then: Case : If p C ε for C = O(, then σ and σ are of o( ε; Case : If C ε < p < C ε, then σ and σ are of O( ε, and σ and σ are slowly varying, i.e. t σ O ( ε, and t σ O ( ε ; Case 3: If p C ε, σ and σ are highly oscillatory with mean. We leave the justification for a simpler model problem to Appendi B. emark 3. The assumption σ mn (t = = (m n is a reasonable assumption. In fact, given arbitrary initial condition, one could check that away from p c, the weak limit of σ mn is always zero, as ε, as can be seen in Appendi B. So numerically we treat the initial data for both σ and σ zero, given the initial velocity S ( away from the crossing point, i.e. σ(,, p δ (p S ( ( a a T, if S ( p c. (33 This assumption is intuitive and empirical, but it does give us some convenience in solving the /S numerically. In fact, the numerical eamples provided in Section 5 indeed show that the band-to-band transition is captured very well with initial data ( A domain decomposition method. Clearly, one has several observations in hand: the classical Liouville is an approimation (in weak limit to away from the crossing point; σ and σ are slowly varying in a neighborhood of p c =, with a small amplitude before the characteristic hitting p c, and a rapid oscillation after that. These observations motivate a domain decomposition method in p-space. The idea is, away from p c, when the classical Liouville equation (7 is a good approimation, we solve this set of equations, and then switch back to when the approimation breaks down around p c (in the O(ε neighborhood of p c. The gain is obvious: numerically it is much easier and more efficient to solve the classical one, thus this approach saves a great amount of computational cost than solving everywhere. Based on the asymptotic properties of the transition coefficients, we propose the following: Given a fied spatial point, the sign of U determines the traveling direction of wave in p. Assume U >, i.e. the wave we study is right-going: Classical regions: p < C ε and p > C ε: In this region, coarse mesh independent on ε is used to solve the classical Liouville system (7, and σ and σ are set to be zero.

13 SEMI-CLASSICAL MODELS FO BAND COSSINGS 3 Semi-classical region: p B \ {Classical region}: Solve. The incoming boundary conditions for σ and σ are set to be zero, and the incoming boundary condition for σ and σ are the inflow boundary condition. A fine mesh is used with and t much less than ε. In case of U <, and the wave if left-going, boundary condition can be set up in the same way. emark 4. Our analysis is based on the regularity of the coefficient matri C in. But usually, the value of C s element can be of O(/δ where δ = min p E (p E (p is the minimal band gap. So C will be large if the minimal band gap δ is small, and the numerical discretization in the semi-classical region should resolve this small parameter δ. In the interested regime ε δ, where the transition cannot be ignored for avoided crossing when E (p c E (p c O( ε, o( ε mesh is enough. 5. Numerical eamples. In this section we solve the and Liouville-S numerically. Firstly in Section 5., we present a numerical method for the equation with a linear eternal potential U(. In this special case, we provide an efficient solver without using the domain decomposition method. In Section 5., the domain decomposition method is applied for general U(. For both eamples, we use the Mathieu model, i.e. the periodic potential is V (z = cos z. The first eight Bloch eigenvalues are shown in Figure. Apparently, th band 5th band E p (a E p (b Figure. The eigenvalues of the Mathieu Model, V ( = cos. some eigenvalues get very close to each other around p =, ±.5. We will focus on the 4th and 5th bands. Denote Ψ and Ψ as the Bloch functions corresponding to the 4th and 5th bands respectively. For comparison, we will compare the numerical results to the ones given by the original equation, computed through the methods given in [7] with mesh size and time step much smaller than ε. The minimum gap between the 4th and 5th bands is.47, located at p =. The gap is considered small enough so that the quantum effect can be seen for the ε being used.

14 4 L. CHAI, S. JIN AND Q. LI 5.. Linear U(. We deal with the linear eternal potential in this section: U( = U β. Then the equation is: [ ( iεφ ε t = ε ] φε + cos + (U β φ ε, (34 ε with the initial data given as a wave packet along the 4th band: ( φ I = a ( Φ ε, S ( e is(/ε, with S ( = p. (35 Correspondingly, the becomes: with given by: t σ + β p σ = σ, σ(,, p = σ I (, p (36a = a ( δ (p S ( ( T, (36b = βc + id ε A. (37 One encounters two computational challenges here. Firstly, one needs to numerically resolve the rapid oscillation. We will present an efficient way to overcome this difficulty by following the characteristics in Section 5... Secondly, the initial data contains a delta function. Usually one uses a Gaussian function with small variance to approimate it, and the error is related to this variance. As will be discussed in Section 5.., in some special cases, this can be avoided by using the singularity decomposition idea of [8]. emark 5. Here we only discuss. Liouville-S can be computed similarly A Fourier transform based integration method. Let the Fourier transform of f(t,, p with respect to be ˆf(t, η, p = e iη f(t,, pd. Taking this transform on (36a, one gets: t ˆσ + β p ˆσ = ( βc + idε iηa ˆσ =: ˆˆσ In this special case, β is a constant so the characteristic line can be obtained analytically. and consequently one can avoid using the mesh thus removes the difficulty due to the high oscillation introduced by the term id ε, as will be clear in the following. We take the first time step t [, t] for eample. Along the characteristic line p(t = p + βt, one evaluate ˆσ and ˆ at (t, p(t, η and has: dˆσ dt = ˆ ˆσ. Solution to this ODE system satisfies ˆσ (t =ˆσ ( + t t iη p E ˆσ (tdt ( β(φˆσ φ ˆσ iη ψ ˆσ dt, (38a

15 SEMI-CLASSICAL MODELS FO BAND COSSINGS 5 and where: and ˆσ (t =e t Kε (τdτ ˆσ ( + t e t s Kε (τdτ ( G(sˆσ (s H(sˆσ (s ds, K ε = i ε (E E iη p E β(φ φ, For t small, an approimation to(38 is G = β φ iηψ, H = β φ. ˆσ (t ˆσ ( iηt p E ( p(t ˆσ (t + ( βφ ( iηψ ( t ˆσ dt βφ ( ˆσ (t ˆσ (e t Kε (τdτ + ( G(ˆσ ( H(ˆσ ( t t e t s Kε (τdτ ds. ˆσ dt, Plug (39b into (39a, to evaluate ˆσ ( t and ˆσ ( t, one needs to compute: F := e t F := F := Kε (τdτ, t t t e t Kε (τdτ dt, e t s Kε (τdτ ds dt. (38b (39a (39b However, to compute F, F and F is not easy since their integrands are highly oscillatory. But if one chooses β t = p, then at each time step one follows eactly the characteristics, so p(t always lie on the grid points, thus F, F and F are time independent, and one only needs to compute them once (with a highly resolved calculation. Similar analysis can be carried out for ˆσ and ˆσ. We prove the stability of this method by using it on a simpler model problem, and it will be justified in Appendi C A singularity decomposition idea. To handle the delta function in the initial condition (36b, one usually approimates it with a Gaussian function with small variance, and numerical error was determined by the width of the Gaussian. As stated before, in some special cases, this error could be avoided, and the eample we are discussing here is for when S ( p = const, for which, we apply the singularity decomposition method introduced in [8] to reduce the error. Write the ansatz of ˆσ mn (t,, p as: σ(t,, p = ω(t,, p δ ( θ(t, p (4 in which: θ(t, p = p (p + βt, which solves the Liouville equation ω satisfies the same equation as σ: t θ + β p θ =, t ω + β p ω = ω. (4

16 6 L. CHAI, S. JIN AND Q. LI These can be proved by simple derivations. Formally, one has σ t = ( ω ω δ(θ = t t δ(θ + ω δ (θ θ t = ω δ(θ β ω p δ(θ βω δ (θ = ω δ(θ β ω p δ(θ βω δ (θ θ p = ω δ(θ β ( ω δ(θ p = σ β σ p. The equalities above should be understood in the distributional sense. The decomposition (4 enables one to solve for ω and θ separately with good (bounded initial data a ( and S ( respectively. The equation ω satisfies is the same as the one for σ, thus the numerical method introduced in Section 5.. can be used. In the final output, one needs to get back to σ using (4, so a discrete delta approimation is only needed at the output time, not during time evolution Numerical eperiments. We show the numerical results of the /S with the following data β =, p 5( π =.5, a ( = ep (. (4 We compute the density, the cumulative density function (c.d.f. and mass in the st band, defined respectively by: ρ ε = φ ε, γ ε = ρ ε (ydy, m (t = P φ ε (t, d, (43 where P n is the projection onto the nth band: P n φ( = dp dyφ(y Ψ n (y, pψ n (, p, φ L (, m N. B The two integrals in (43 are calculated by the midpoint quadrature rule numerically. Figure shows the density and c.d.f. computed for the equation, the and the Liouville-S respectively at t =.5. The results match quite well. Figure 3 shows the evolution of m as a function of time t. One can see the total mass on the first band jumps down at around t =.5, when the momentum p reaches p c =, reflecting the 4th-to-5th band transition. The eperiment also shows that smaller ε gives smaller transition rate. Note that some small oscillations occur around the crossing region. They are related to the interference phenomena, and are usually called the Stueckelberg oscillation [8, 3, 33]. Define L error in the cumulative distribution function (c.d.f. [4, ]: Err ε ( (t = ρ ε S (t, z ρ ε L(t, z dz d, (44 where ρ ε S and ρε L denote the density calculated by the equation and the Liouville system respectively. Numerically we compute (44 using the midpoint

17 SEMI-CLASSICAL MODELS FO BAND COSSINGS Liouville-S.8 Liouville-S ρ ε γ ε (a ε = Liouville-S.8 Liouville-S ρ ε.5.5 γ ε (b ε = Liouville-S.8 Liouville-S ρ ε.5.5 γ ε (c ε = Figure. The eample in Sec. 5..3, t =.5. The left and right column are for the position density ρ ε, and c.d.f. γ ε respectively. The solid line, the dash line and the dotted line are the numerical solutions to the, the and the Liouville-S respectively.

18 8 L. CHAI, S. JIN AND Q. LI.8 Liouville-S.8 Liouville-S.6.6 m.4 m t (a ε = t (b ε = 9.8 Liouville-S.9.8 Liouville-S m.6.4 m t (c ε = t (d ε = Figure 3. The eample in Sec : time evolution of m (t defined in (43. quadrature rule. Figure 4 shows this at time t =.5. As ε, the Liouville system gets more accurate, and the error decreases with the speed of O(ε. 5.. A domain decomposition computation. This section shows eamples with varying U. For this general case, the p-characteristic is no longer a straight line, and the fast solver in the previous section is no longer valid. To numerically solve, we use the domain decomposition method. The classical finite volume method is used for the convection terms. We compute the system with both a pure and a mied state initial data with: U( = sin A pure state initial data. In this eample, we use the same pure state initial data as in the previous eample (35,(4. Correspondingly, the initial data for the system is given by (36b. Numerically a Gaussian function centered at p with variance of ε/6 is used to approimate the δ function. Figures 5 and 6 show the density, the c.d.f at t =.5 and evolution of m (43 computed for both the equation and the system. The numerical results for the two systems agree well. Figure 7 gives decay of Err ε (44.

19 SEMI-CLASSICAL MODELS FO BAND COSSINGS 9.5. Liouville-S.5 Err ε ε Figure 4. The eample in Sec : Err ε as function of ε at t =.5. The numerical results show that the hybrid model can capture the band-to-band transition phenomena, and the error decays like O(ε. emark 6. As U varies with, the wave packet becomes decoherent. This weakens the interference phenomenon [8, 3, 33]. As one can see in Figure 6, the Stueckelberg oscillations around the crossing region is much weaker than those in the previous eample A mied state initial data. This eample is for the case when the initial data is a mied state: [ ( ( ] = a ( Φ ε, p e ip /ε + Φ ε, p e ip /ε, p =.5. φ I Correspondingly, the initial data for the semi-classical Liouville system should be: σ = a ( δ(p p [,,, ] T. Since p is away from the crossing point and σ and σ weakly converge to zero as ε, numerically, we regard them as zero and use σ a ( δ G (p p [,,, ] T as the initial condition, where δ G (p is a Gaussian function centered at zero. The density and the c.d.f. are computed for the and the, compared in Figure 8. Err ε as a function of ε is shown in Figure Conclusion. In this paper we derive semi-classical models for the linear equation with periodic potentials. These models take into account the band crossing, which is important to describe quantum transitions between different Bloch bands. Away from the band-crossing zones these models reduce (in the sense of weak limit to the classical Liouville system for each Bloch band. We also couple these semi-classical models (to be used near the crossing zones and the classical

20 L. CHAI, S. JIN AND Q. LI ρ ε γ ε (a ε = ρ ε γ ε (b ε = ρ ε.5 γ ε (c ε = Figure 5. The eample in Sec t =.5. The left and right columns show the position density ρ ε, and the c.d.f. γ ε respectively. Liouville equation (used away from the crossing zones for an efficient multiscale computation. Our numerical eperiments show that these semi-classical models provide correct quantum transitions near the crossing zones when compared with the direct simulation of the original equation.

21 m.6 m t (a ε = t (b ε = m.8.6 m t (c ε = t (d ε = Figure 6. The eample in Sec. defined in ( : time evolution of m (t Appendices A. Hyperbolicity of the two-band semi-classical Liouville systems. The semi-classical Liouville system ( is where t σ + A σ U p σ = Sσ (45 S = C + id ε, and σ, A, C, D are defined in (. Noted that A = A and S = S. To check the hyperbolicity, we separate the real and imaginary parts. σ = e σ + i Im σ, A = e A + i Im A, S = e S + i Im S, with e F and Im F denoting the real part and imaginary part of F respectively.

22 .6.4. Err ε ε Figure 7. The numerical eample in Sec. 5..: the L error Err ε between and the solution at t =.5. Define σ I = ( e σ, Im σ T, and [ ] [ e A Im A e S Im S A I =, and S Im A e A I = Im S e S ], then one can get t σ I + A I σ I U p σ I = S I σ I. (46 Since A = A, then e A = e A T, Im A = Im A T. Thus have that the matri A I = A T I is a symmetric matri, which implies the hyperbolicity of the system. In addition, since S = S, then e S = e S T, Im S = Im S T. Therefore the matri S I = SI T is a skew-symmetric matri. Similarly, one can obtain the hyperbolicity of system Liouvill-S (3. B. Some basic analysis of the semi-classical Liouville systems. To understand the asymptotic behavior of the solution to the system (, as mentioned in Section 4., we look at a simpler model system: t g + f + b( p g =, t f + a(p f + g + b( p f = i ε c(pf, (47 g(,, p = g I (, p, f(,, p = f I (, p. The initial conditions g I and f I are bounded smooth functions independent on ε, b > and the set of zeros for c(p: S c = {p : c(p = } is measured zero. It is easy to check that (47 is a linear hyperbolic system, and the solutions g and f are bounded uniformly in ε [9].

23 ρ ε γ ε (a ε = ρ ε γ ε (b ε = ρ ε γ ε (c ε = Figure 8. The eample in Sec. 5..: at time t =.5. The left and the right columns show the position density ρ ε, and the c.d.f. γ ε respectively. B.. Weak convergence. We consider the weak limit of the solution of (47 in this subsection. To do this, we introduce the inner product, as u, v = u(t,, p v(t,, p d dp dt.

24 Err ε ε Figure 9. The numerical eample in Sec. 5..: the L error Err ε as a function of ε at t =.5. Choose an arbitrary test function h C ( +, take the inner product on both side of (47 w.r.t h, one gets { t g, h f, h + b p g, h =, f, t h + af, h + g, h + bf, p h = i ε cf, h. (48 The derivatives in the equation of (48 are acted on the smooth function h, and the left side is bounded. One gets cf, h as ε for all h C ( +. Given that c is almost surely nonzero, and f is bounded, one gets f weakly. Combined with the first equation in (48, one gets t g + b( p g weakly. B.. Strong convergence: for constant b. In these two subsections, we formally prove that before getting close to the crossing region, c(p is assumed to be bigger than a constant c that is unrelated to ε. In this region, f is constantly small and controlled by O(ε. This subsection is for the case when the speed on p direction is a constant: b( = β. Along the p-characteristic line p(t = p + βt, one applies the Fourier transform to the -variable, and gets: where f(t, η = ( ĝ(t, η, p(t, ˆf(t, η, p(t T and ( η (t = η c(p(t/ε ηa(p(t d f = i(tf, (49 dt.

25 5 The two eigenvalues of (t are both real, and thus the system above has a bounded solution satisfying The equivalence between norms gives: ĝ(t, η + ˆf(t, η = ĝ I (η + ˆf I (η. ĝ(t, η + ˆf(t, η < C( ĝ I (η + ˆf I (η. Adopt it into the solution to (49, one gets t g(t, = π η ˆf(t, ηe iη dη η π ˆf(t, η dη ( C η ĝ I (η + ˆf I (η dη t g(t, = π η ˆf(t, ηe iη dη η π ˆf(t, η dη ( C η ĝ I (η + ˆf I (η dη If the initial conditions ĝ I and ˆf I are smooth enough, and decay fast as η, one could easily get that t g(t, and tg(t, are both bounded in time independent of ε, and thus g and g are slowly varying in time. emark 7. In the derivation above, we dropped the p(t-dependence in the functions for simplicity. The partial derivative in the t-variable t should be understood as taking along the p-characteristic line p(t, i.e. t g(t, = t g(t,, p + β p g(t,, p. Assume that f I (, one follows the characteristics of (t by solving ẋ(t = a(t, and gets: f ( t, (t, p(t t ( i t g ( = ep c (p(τ dτ t s, (t s, p(t s ds. ε t s By the assumption, before hitting the crossing region, c(p(τ > c >, then the stationary phase argument suggests that, given slowly varying g(t,, p(t, f = O(ε. The observations from the above two subsections suggest that f, and before getting close to the crossing region, f is as small as of O(ε. Based on these arguments, for the system (, and propose the following conjecture: if σ and σ are initially zero, then: Case : If p ε, then σ and σ are of o( ε; Case : If p [ ε, ε], then σ and σ are of O( ε, and slowly varying; Case 3: If p ε, σ and σ are highly oscillatory, and converge to weakly. C. The integration method of a simple model system. In this section, we apply the method in (39 onto a simple model to show stability. d dt f(t, p + t = (p + t f(t, p + t, where f(t, p = (g(t, p, f(t, p T, p = p + t, and ( r (p r (p = (p r (p r(p ε.

26 6 with r(p ε = r (p + i ε c(p, while r and r are purely imaginary, and c(p real and positive. r, r and r are independent on ε. Set up mesh as t j = j t, and p i = + (i p, with p and t being the mesh size. Denote g j i and f j i as the numerical result at (t j, p i, then (39 gives f(t, p(t = f j i e t rε (pi +τdτ g j i r (p i t g(t, p(t = g j i + g(t, p(t r (p(t + r (p i e t s rε (pi +τdτ ds, t (5a f(t j + s, p i + s ds (5b Plug (5a into (5b, and evaluate them at (t j+, p i+, one obtains: ( r (p i+ t ( t t g j+ i+ = g j i r (p i e t s rε (pi +τdτ ds dt Written in vector form gives with M i = +f j i r (p i t ( r(p i t t e ts r ε (pi +τdτ ds dt r (p i+ t e t rε (pi +τdτ dt. f j+ i+ = M i f j i, (5 r (p i t e t r ε (pi +τdτ dt r (p i+ t r (p i t e t s r ε (pi +τdτ ds e t r ε (pi +τdτ The following quantities in the matri M i should be evaluated very accurately: F = e t F = F = r ε (pi +τdτ, t t t e t rε (pi +τdτ dt, e t s rε (pi +τdτ ds dt. emark 8. These three quantities only depend on the mesh grid point inde i but not the time steps inde j, thus they only need to be computed once at the beginning of the computation. Given ε t, the integrands of F and F are highly oscillatory, and one can see that F O(ε and F O(ε. Simple calculation shows that M i can be written as M i = Ω M i, with ( ( Ω = diag r (p j+ t, r (p, Mi = i F r F. r F F With purely imaginary r and r ε, it is easy to prove that Ω, and M i ( + O(ε t, and thus M i ( + O(ε t. This implies asymptotic stability of the scheme (5 independent of ε..

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