Adiabatic Evolution Through Infinitely many Eigenvalue Crossings

A. Joye^{∗}, F. Monti^{†}^{‡}, S. Guérin^{†}, H.R. Jauslin^{†}

Abstract

We prove an Adiabatic Theorem adapted to models that present an instantaneous eigenvalue experiencing an infinite number of crossings with the rest of the spectrum. Our formulation is adapted to the study of the dynamics of molecules interacting with a laser field in the Adiabatic Floquet formalism in presence of frequency chirping.

1 Introduction

1.1 Physical Setting

The availability of intense pulsed laser sources has opened a large field of possibilities to control atomic and molecular dynamical processes. One of the main theoretical tools to analyze these processes is adiabatic Floquet theory, [8] and references therein. In the general setup, a molecule, described by a Hamiltonian H_{0} acting on a Hilbert space H, interacts with one radiation mode of frequency ω. The field is treated as a classical field. The Hamiltonian of the total system can be written, for example, as

H(ω t + θ_{0}) = H_{0} + E M F (ω t + θ_{0})  (1)

where M is the dipole moment operator, of the molecule, E ∈ ℝ is a parameter representing the amplitude of the radiation field, F is a real valued 2π-periodic function and θ_{0} the initial phase. We assume that H_{0} has a discrete spectrum. In order to describe a laser pulse the amplitude is taken as a slowly varying time dependent function E(ε t). As it has been shown in [6] [7], the possibility to vary the frequency is a powerful method to enhance the control of molecular processes driven by lasers. This frequency chirping is modelled by modulating slowly the frequency ω = ω(ε t) within the pulse duration.

The Hamiltonian of the model has two kinds of time-dependencies. The periodic part is treated by Floquet methods, and the slowly varying part by adiabatic theory. Adiabatic Floquet theory is based on the following statement: Assume that in the Hamiltonian (1) the parameter E and the frequency ω are made time dependent, E(t), ω(t), and M stays time independent. Consider the propagator U(t, t_{0}; θ_{0}), solution of the Schrödinger equation

∗Institut Fourier, Unité Mixte de Recherche CNRS-UJF 5582, Université de Grenoble I, BP 74, 38402 Saint Martin d’Hères Cedex, France

†Laboratoire de Physique, CNRS, Université de Bourgogne, BP 400, 21011 Dijon, France

‡École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland

i \frac{∂}{∂t}U(t, t_{0}; θ_{0}) = H(ω t + θ_{0}) U(t, t_{0}; θ_{0}),  U(t, t; θ_{0}) = 1l  (2)

acting on the Hilbert space H. Tensoring H with the space of square integrable functions on the unit circle: L^{2}(𝕊^{1}, H) we get an enlarged Hilbert space in which the operator U(t, t_{0}; θ) can be lifted by interpreting the θ-dependence as a multiplication operator. We can then define

U_{K}(t, t_{0}) = e^{−tω(t)∂_{θ}} U(t, t_{0}; θ) e^{t_{0}ω(t_{0})∂_{θ}}.

The statement is that equation (2) is equivalent to

i \frac{∂}{∂t}U_{K}(t, t_{0}) = K(t) U_{K}(t, t_{0})  (3)

with

K(t) = −i ϖ(t) ∂_{θ} + H_{0} + E(t) M F (θ)

and ϖ(t) denotes an effective instantaneous frequency defined by ϖ(t) = ω(t)+t dω(t)/dt. If the time dependence of E(t), ω(t) is slow, we can apply adiabatic techniques for the evolution of (3). When K has pure point spectrum, the instantaneous eigenvalues and eigenvectors can be written and labelled in the form

λ_{j,k} = λ_{j,0} + ϖ k,  k ∈ ℤ  (4)
ψ_{j,k}(θ) = ψ_{j,0}(θ) e^{i k θ}.

The index j has the same cardinality as the dimension of the Hilbert space H. As functions of E and ϖ, these eigenvalues may exhibit crossings, which the adiabatic approximation can accomodate in case there is a finite number of them, see [5], [9]. The structure (4) of the eigenvalues is such that if we consider a slowly varying effective frequency ϖ(t) that goes through 0 at some time t_{0}, the nature of the spectrum becomes quite different. One can encounter situations in which a branch of instantaneous eigenvalues undergoes an infinite number of crossings with other branches, or the spectrum may become suddenly continuous. Note that a linear decrease of the (strictly positive) frequency ω(t), naturally gives rise to an effective frequency ϖ(t) that goes through zero.

Hence it becomes necessary to investigate the validity of the adiabatic theorem in such situations. We will consider the case where a branch of eigenvalue undergoes an infinity of crossings with other branches. We formulate an adiabatic theorem that can be applied to such situations with an estimate on the corrections to the adiabatic limit. We also provide a whole class of systems for which this situation is true.

1.2 Adiabatic Theorems

We refer the reader to the recent surveys [2], [12] and references therein for a detailed account on the adiabatic Theorem. We simply recall here that under the usual gap assumption we have a quite general formulation of the adiabatic Theorem where the error term is well controlled and of order ε, see e.g. [4].

In case the gap assumption is modified, the situation is less explicit. We switch back to the notation H(ε t) for the slowly varying time-dependent Hamiltonian. Assume H(s) is smooth and there exists a spectral projector P (s) of H(s) which is strongly C^{2}. Then the adiabatic theorem holds under these conditions, provided P (s) is of finite rank, independently of any spectral considerations , see [1]. The limitation of this approach is that, in general, no estimate can be made on the rate at which the adiabatic regime is attained. However, in certain specific situations an estimate on this rate is available. It the case for example if the spectral measure μ_{φ} is α-Hölder continuous, with φ = P′(s)ψ, ψ being the initial condition [1]. A case where the spectrum of H(s) is dense pure point is dealt with in [3]. Another situation where the gap hypothesis is not necessarily fulfilled occurs when H(s) = H_{0}(s) + εH_{1}(s), where the domain of H_{1}(s) is smaller than that of H_{0}(s), see [10].

We report here on another situation in which the usual gap assumption is modified and the error made in the adiabatic approximation can be estimated. Roughly speaking, we consider a spectral projector P (s) associated with an eigenvalue λ(s), s ∈ [0, 1]. λ(s) is isolated in the spectrum except at a series of times {o_{k}}_{k∈ℕ} accumulating at a ∈]0, 1[ where it experiences crossings with the rest of the spectrum. Under some conditions on the local behaviour of the gap between λ(s) and the rest of the spectrum near the crossing points o_{k}, we estimate the error term in the theorem without a priori knowledge on the nature of the rest of the spectrum.

2 Main Result

Regularity hypothesis: H0) For all s ∈ [0, 1], H(s) is a bounded strongly C^{1} self-adjoint operator in a separable Hilbert space K. There exists of a spectral projector P (s) of H(s) which is strongly C^{2} on [0, 1] and such that H(s) P (s) = P (s) H(s) = λ(s) P (s), for all s ∈ [0, 1].

Note that in order to treat the application described above, it is necessary to consider unbounded hamiltonians, with time dependent domains. This causes technical difficulties only that are dealt with in [11].

Consider first one crossing of λ with the rest of the spectrum. Let g(s) be the gap between λ(s) and the rest of the spectrum of H(s): g(s) = dist (λ(s), σ(s)\{λ(s)}) ≥ 0, s ∈ [0, 1] and L(s) = i [P′(s), P (s)]. Assume that g^{−1}{0} = {o} and consider the equations

iε U′(s) = H(s) U(s) ,  U(0) = 1l  (5)
iε A′(s) = (H(s) + εL(s)) A(s) ,  A(0) = 1l .  (6)

The unitary A is the adiabatic evolution which possesses the well known intertwining relation A(s) P (0) = P (s) A(s) [13]. Let W (s) be defined by W (s) = A^{−1}(s) U(s). We have

iW′(s) = −A^{−1}(s)L(s)A(s) W (s),  W (0) = 1l .  (7)

To compare the adiabatic and actual evolutions, we estimate the difference between W (s) taken at two times surrounding the crossing:

Lemma 2.1 For any 0 ≤ u_{0} ≤ t < o < s ≤ u_{1} ≤ 1,

∥W (u_{0}) − W (u_{1})∥ ≤ C (ε|u_{0} − t|/g_{t}^{2} + ε|u_{1} − s|/g_{s}^{2} + ε/g_{t} + ε/g_{s} + |s − t|) (8)

where g_{t} = inf_{u∈[u_{0},t]} g(u), g_{s} = inf_{u∈[s,u_{1}]} g(u) and the constant C is uniform in u_{0}, u_{1}, s and t.

Note that the classical paper by Born and Fock, [5], and the detailed analysis of crossings by Hagedorn [9] provide sharper estimates with more informations on the spectrum.

To deal with an infinite number of crossings, we assume that the crossings points are well separated and that the gap behaves as power law of order at most α. Moreover, outside some local crossing region the gap must relatively large. This means that the only accumulation point of small gaps is a. More precisely:

Spectral hypothesis: There exist two partitions {u_{k}^{±}}_{k∈ℕ} of [0, a) and (a, 1] respectively:

0 = u_{0}^{−} < … < u_{k−1}^{−} < u_{k}^{−} … ⟶ u_{∞}^{−} = a = u_{∞}^{+} ⟵ … u_{k}^{+} < u_{k−1}^{+} < … < u_{0}^{+} = 1

such that for each k ∈ ℕ^{∗}, H1) one can find non empty open intervals V_{k}^{±}, which satisfy V_{k}^{−} ⊂ [u_{k−1}^{−}, u_{k}^{−}], V_{k}^{+} ⊂ [u_{k}^{+}, u_{k−1}^{+}] and

sup_{s∈V_{k}^{±}} g(s) ≤ inf_{t∈I_{k}^{±}} g(t)  (9)

where I_{k}^{−} = [u_{k−1}^{−}, u_{k}^{−}]\V_{k}^{−} and I_{k}^{+} = [u_{k}^{+}, u_{k−1}^{+}]\V_{k}^{+}. H2) there are constants G_{±}(k) > 0 and a k-independent positive constant α such that for all s ∈ V_{k}^{±}:

G_{±}(k) |s − o_{k}^{±}|^{α} ≤ g(s) ,  (10)

for some points o_{k}^{±} ∈ V_{k}^{±}.

To estimate the difference between the real evolution and the adiabatic one, we apply Lemma 2.1 on a finite number of crossings and take a simple integral bound over the rest of the interval surrounding a. The choice of the number of crossings will be optimized with respect to ε in order to get a good bound of the remainder term.

Let F_{k} ∼ f(k) mean that there exist two constants 0 < c_{1} < c_{2} < ∞ such that c_{1} f(k) ≤ F_{k} ≤ c_{2} f(k) for k ∈ ℕ^{∗} large enough. Our main result reads:

Proposition 2.1 Assume the above hypothesese and the following behaviours:

|u_{k}^{±} − a| = C_{1}/k^{β} + C_{2}/k^{β+1} + o(1/k^{β+1})) ,  β > 0, C_{1} ≠ 0
G_{±}(k) ∼ k^{γ} ,
|V_{k}^{±}| ∼ 1/k^{δ} ,  δ > 0 .

We set μ = min{β + 1 + 2γ, α(β + 1) + γ}. Then ∥U(1) − A(1)∥ = O(ε^{p}) where the exponent p is given by

p = (1 + 2 α)^{−1} if μ > (1 + 2α)

p = (1 + 2 α)^{−1} − ν ∀ν > 0 if μ = (1 + 2α)
p = β((β + 1) (1 + 2α) − μ)^{−1} if μ < (1 + 2α)

provided δ satisfy the constraints: β + 1 ≤ δ ≤ β + max{1, μ/(1 + 2α)}.

Remarks:

i) This Proposition actually derives from a more general Theorem proven in [11]. ii) We show below that the values α = β = γ = 1 and δ = 2 are generic in some sense. For these values the exponent p = 1/3 − ν, for all ν > 0 can be improved by a more careful analysis to p = 1/3.

3 Application

We now consider a family of models for which the situation just described takes place as the effective frequency ϖ takes the value zero. The most general two level system driven by a periodic field is characterized by choosing freely the eigenvalues λ_{+,m} = λ_{+} + m ω and λ_{−,k} = λ_{−} + k ω and the corresponding eigenfunctions:

ψ_{+,m}(θ) = ( \begin{matrix}e^{i x(θ)} cos z(θ) \\ e^{i y(θ)} sin z(θ) \end{matrix} ) e^{i m θ}  and ψ_{−,k}(θ) = ( \begin{matrix}−e^{−i y(θ)} sin z(θ) \\ e^{−i x(θ)} cos z(θ) \end{matrix} ) e^{i k θ} . (11)

The functions x, y and z are periodic modulo integer multiple of θ. With the matrix

Υ(θ) = ( \begin{matrix}e^{i x(θ)} cos z(θ) & −e^{−i y(θ)} sin z(θ) \\ e^{i y(θ)} sin z(θ) & e^{−i x(θ)} cos z(θ) \end{matrix} ) ,

the corresponding Floquet Hamiltonian can be written as

K = −i ϖ ∂_{θ} − i ϖ Υ(θ) (∂_{θ}Υ^{−1}(θ)) + Υ(θ) D Υ^{−1}(θ)  (12)

where D = diag(λ_{+}, λ_{−}). Introduce now a supplementary smooth slow time dependence in the parameter x = x(θ, s), y = y(θ, s), z = z(θ, s), λ_{±} = λ_{±}(s) and ϖ = ϖ(s). The eigenvalues, eigenvectors and the corresponding eigenprojectors become smooth functions of s, so that the regularity Hypothesis H0) is satisfied. Note that when ϖ = 0 the operator K reduces to the (matrix) multiplication operator by Υ(θ) D Υ^{−1}(θ) on L^{2}(𝕊^{1}, ℂ^{2}), whose spectrum consists of two eigenvalues λ_{±} which are infinitely degenerate. In the general situation, if ϖ = 0, K becomes a multiplication operator by an arbitrary 2π periodic 2 × 2 matrix H(θ). In that case, the spectrum of K is continuous and given by two band functions which are the instantaneous (in θ) eigenvalues of H(θ).

We assume that λ_{+}(s) = −λ_{−}(s) ≡ η(s)/2 > 0 and, for simplicity, we take ϖ(s) = s. We select the eigenvalue λ(s) = λ_{+,0}(s) and denote by ψ(s) the associated eigenvector such that P (s) = |ψ(s)⟩⟨ψ(s)|. The only crossings that λ experiences are with the eigenvalues λ_{−,k}’s and they take place at times s such that k s = η(s), k ∈ ℕ^{∗}. The next Lemma shows that, generically, the spectral hypothesese of Proposition 2.1 are satisfied for the models (12). We focus on the interval (0, S], for S small enough, the interval [−S, 0) being dealt with similarly. We also drop the + sub/super-scripts.

Lemma 3.1 Assume the eigenvalues can be written as λ_{±,m}(s) = m s ± η(s)/2, where η is a C^{2} function with bounded derivatives such that η(0) > 0. Then

i) the crossing points are given by o_{k}, for k ∈ ℕ large enough, such that

o_{k} = η(0)/(k − η′(0)) + O(1/k^{3}).  (13)

ii) the sequence u_{k} > 0 defined by η(u_{k}) = (k + 1/2) u_{k} is such that for k ∈ ℕ large enough, u_{k} < o_{k} < u_{k−1} and

u_{k} = η(0)/(k + 1/2 − η′(0)) + O(1/k^{3}).  (14)

iii) there exists r_{k} ∈ (o_{k}, u_{k−1}) such that the sets V_{k} = (u_{k}, r_{k}) are disjoint, |V_{k}| ∼ 1/k^{2}, and H1 is satisfied with I_{k} = {u_{k}} ∪ [r_{k}, u_{k−1}]. iv) For each s ∈ V_{k}\{o_{k}}, H2 holds as well and

g(s) = |η(s) − k s| ∼ k |s − o_{k}| .  (15)

Hence we get α = 1, β = 1, γ = 1 and δ = 2. Thus, the adiabatic approximation holds for the models (12) with an error term O(ε^{p}), p < \frac{1}{3}. In keeping with the last remark of Section 2, we recall that a more careful analysis yields p = 1/3.

References

[1] J.E. Avron, A. Elgart, Commun. Math. Phys. 203, 445-463, (1999)

[2] J.E. Avron, A. Elgart, To appear in Proceedings of Qmath7, Prague 1998

[3] J. Avron , J. Howland, B. Simon, Commun. Math. Phys. 128, 497-507 (1990).

[4] J.E. Avron, R. Seiler, L.G. Yaffe, Commun. Math. Phys. 110, 33–49 (1987).

[5] M. Born, V. Fock, Zeit. f. Physik 51, 165–180 (1928).

[6] S. Chelkowski, A.D. Bandrauk, P.B. Corkum, Phys. Rev. Lett. 65, 2355-2358 (1990).

[7] S. Guérin, Phys. Rev. A 56, 1458-1462 (1997).

[8] S. Guérin, F. Monti, J.-M. Dupont, H.R. Jauslin, J. Phys. A: Math. Gen. 30, 7193–7215 (1997).

[9] G. Hagedorn, Ann. Phys. 196, 278-295 (1989).

[10] A. Joye, Ann. Inst. H. Poincaré, Sect. A 63, 231–250 (1995).

[11] A. Joye, F. Monti, S. Guérin, H.R. Jauslin, preprint 1998.

[12] A. Joye, Ch.-Ed.Pfister, ICMP ’97, De Wit, Bracken, Gould and Pearce Eds, International Press, Cambridge, pp 309-315, 1999

[13] T. Kato, J. Phys. Soc. Japan 5, 435-439 (1950).