Exponential Estimates in Adiabatic Quantum Evolution^{1}

\underline{Alain Joye}^{2} and Charles-Edouard Pfister^{3}

Abstract

We review recent results concerning the exponential behaviour of transition probabilities across a gap in the adiabatic limit of the time-dependent Schrödinger equation. They range from an exponential estimate in quite general situations to asymptotic Landau-Zener type formulae for finite dimensional systems, or systems reducible to this case.

1 Introduction

The notion of adiabatic evolution or adiabatic process is an important theoretical concept, which occurs at several places in Physics. In Quantum Mechanics, this process is usually described by the equation ih̄\frac{∂}{∂t′} ψ(t′) = H(εt′)ψ(t′), where ε is a small parameter, such that 1/ε gives the typical time-scale over which the Hamiltonian changes significantly. Setting h̄ = 1 and introducing a rescaled time, t = εt′, we can rewrite the time-dependent Schrödinger equation for the evolution operator U as

iε\frac{∂}{∂t}U(t, s) = H(t)U(t, s) , U(s, s) = I , ∀t, s ∈ R. (1)

The adiabatic limit corresponds to the singular limit ε → 0 of the equation (1). If the state of the system is an eigenfunction ψ(t_{0}) for the eigenvalue e(t_{0}) at t = t_{0}, then, in the adiabatic limit, the state of the system at time t = t_{1} is an eigenfunction ψ(t_{1}) for the eigenvalue e(t_{1}), provided the energy-level e(t) is isolated in the spectrum of the Hamiltonian H(t) for all t in the time-interval [t_{0}, t_{1}] [4]. When the system performs a cycle, H(t_{0}) = H(t_{1}), and the energy-level e(t) is non-degenerate, the eigenfunction ψ(t_{1}) defined in the adiabatic limit differs from ψ(t_{0}) by a phase, which can be decomposed into an ε-dependent dynamical phase and an ε-independent geometrical phase related to the spectral subspaces visited during the adiabatic evolution. This is the fundamental observation of Berry, which gave rise to extensive developments (see the collection of papers in [32]). In this note we review a complementary aspect, namely the estimation of the probability that the final state of the system is not an eigenstate ψ(t_{1}) for e(t_{1}); such a transition is called adiabatic or nonadiabatic transition in the physical literature. There are two kinds of results. On the one hand for small systems (or systems reducible to this case), typically two-level systems, one can derive explicit formulae for the probability of such a transition. A particular case is the Landau-Zener formula used when the two levels display an avoided crossing. For general systems, on the other hand, one can usually obtain upper estimates for the transition probability only. Those estimates are reviewed in the second part. The whole discussion is lead in a scattering setting under the assumption that the time-dependence of the Hamiltonian is analytic. More precisely H1 The self-adjoint family {H(t)}_{t∈R} is defined on a common dense domain D of a separable Hilbert space H; it is uniformly bounded from below; for each ϕ ∈ D, H(t)ϕ has an analytic extension in a fixed open strip S ⊂ C including the real axis. Moreover, there exist H(±∞), two self-adjoint operators defined on D, and constants C, α > 0 such that for t{> \atop <}0, sup_{s | t+is∈S} ∥(H(t + is) − H(±∞))φ∥ ≤ C \frac{∥φ∥+∥H(±∞)φ∥}{(1+|t|)^{1+α}}. H2 The spectrum σ(t) of H(t) consists in two disjoint parts σ(t) = σ_{1}(t) ∪ σ_{2}(t), such that inf_{t∈R} dist (σ_{1}(t), σ_{2}(t)) ≥ g > 0 and σ_{1}(t) is bounded. Denoting by P_{j}(t) the spectral projector associated with σ_{j}(t), j = 1, 2, the transition probability between the subspaces P_{1}(t_{0})H et P_{2}(t_{1})H is defined as P(t_{1}, t_{0}, ε) = ∥P_{2}(t_{1})U(t_{1}, t_{0})P_{1}(t_{0})∥^{2}. The Adiabatic Theorem of Quantum Mechanics [4], [21], [28],

1Proceedings of XIIth International Congress of Mathematical Physics, Brisbane, Australia, 1997.

2Institut Fourier, Université Grenoble-1, B.P. 74, F-38402 Saint-Martin d’Hères

3Département de Mathématiques, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne

[1] states that P(t_{1}, t_{0}, ε) = O(ε^{2}) as soon as H(t) is C^{3}. Under the assumption H1 (analyticity and existence of H(±∞)) we have P(+∞, −∞, ε) = O(ε^{∞}) [25], [29], [22]. The difficult questions are to prove exponential estimates and asymptotic formulae for P(+∞, −∞, ε).

2 Asymptotic formulae

Assume the Hamiltonian H(t) is an n×n hermitian matrix with non degenerate spectrum σ(t) = {e_{1}(t), ⋯, e_{n}(t)}. We determine up to a global phase factor a corresponding basis of normalized eigenvectors φ_{j}(t), j = 1, ⋯, n, by requiring that ⟨φ_{j}(t)|φ′_{j}(t)⟩ ≡ 0, t ∈ R. We can expand the solution ψ(t) = U(t)ψ(0) of (1) as

ψ(t) = ∑_{j=1}^{n} c_{j}(t)e^{−i ∫_{0}^{t} e_{j}(t′)dt′/ε}φ_{j}(t), (2)

where the c′_{j}s are complex valued coefficients and the phase factors e^{−i ∫_{0}^{t} e_{j}(t′)dt′/ε} are introduced for convenience. It follows from H1 that the limits lim_{t→±∞} c_{j}(t) = c_{j}(±∞) exist. Thus, choosing c_{j}(−∞) = δ_{jk} as initial conditions, we get that |c_{j}(∞)|^{2} yields the transition probability from the eigenspace associated with e_{k}(−∞) to the eigenspace associated with e_{j}(+∞), which we denote by P_{jk}(ε). Born and Fock [4] showed that

c_{j}(t) = c_{j}(−∞) + O(ε) (3)

uniformly in t ∈ R, from which the Adiabatic Theorem P_{jk}(ε) = O(ε^{2}) follows.

In case n = 2, for real symmetric analytic Hamiltonians, the pioneering works [24], [34] and [5] gave arguments in favour of the Landau-Zener-Dykhne formula P_{12}(ε) ≃ e^{−2γ/ε} with γ > 0 explicited below. A convincing derivation of this formula based on the integration of (1) in the complex t-plane can be found in the important contribution [9]. However, for complex hermitian matrices, this formula misses a prefactor of geometrical nature, as we now show. The proof is based on the fact that the solution ψ of (1) is analytic throughout the strip S, since H is, whereas φ_{j} and e_{j} have multivalued extensions. More precisely, their only possible branching points are the complex crossing points z_{0} ∈ S, such that e_{1}(z_{0}) = e_{2}(z_{0}). Generically, e_{2}(z) − e_{1}(z) ≃ \sqrt{z − z_{0}}. Let η_{0} be a negatively oriented loop based at the origin, which encircles z_{0}. Denoting by f(z|η_{0}) the analytic continuation along η_{0} of a function f(z) defined in a neighbourhood of the origin, we can write e_{1}(z|η_{0}) = e_{2}(z). It follows that φ_{1}(z|η_{0}) is proportional to φ_{2}(z) and we define θ_{21}(η_{0}) ∈ C by φ_{1}(z|η_{0}) = e^{−iθ_{21}(η_{0})}φ_{2}(z). Since ψ(z|η_{0}) = ψ(z), we deduce from the foregoing and (2) the key identity

c_{1}(z|η_{0}) = e^{iθ_{21}(η_{0})}e^{i ∫_{η_{0}} e_{1}(z′)dz′/ε}c_{2}(z). (4)

Let us assume, for simplicity, that there exists a unique generic crossing point z_{0} in S with Im z_{0} > 0. By analyticity of ψ, we can integrate (1) from −∞ to +∞ with c_{j}(−∞) = δ_{1j} along the real axis or along any path β ⊂ S which passes above z_{0}. Denoting by c̃_{1} the analytic continuation of c_{1} along such a path β, we have, due to (4), c_{2}(+∞) = e^{−iθ_{21}(η_{0})}e^{−i ∫_{η_{0}} e_{1}(z′)dz′/ε}c̃_{1}(+∞). It thus remains to prove the estimate c̃_{1}(+∞) = 1 + O(ε), which is the equivalent along β of the estimate (3). As is well known from complex WKB methods, such an estimate can be proven under a global dissipativity condition on the path β only. By definition, a path β ⊂ S parametrized by t ↦ β(t) with lim_{t±∞} Re β(t) = ±∞ is called dissipative (for the indices {1, 2}) if Im ∫_{β(t)}(e_{1}(z) − e_{2}(z))dz is non-decreasing in t ∈ R, where ∫_{β(t)} means integration from −∞ to β(t) along β(s), −∞ ≤ s ≤ t. Sumerizing the above discussion we have Theorem 1 [12] Assume H1, H2 for H(t) a 2 × 2 matrix and suppose there exists a unique generic crossing point z_{0} in S with Im z_{0} > 0. Then, provided there exists a dissipative path in S going from −∞ to +∞ above z_{0}, we have for ε > 0 small enough

P_{21}(ε) = e^{2Im θ_{21}(η_{0})}e^{2Im ∫_{η_{0}} e_{1}(z)dz/ε}(1 + O(ε)). (5)

Remarks: The complex factor e^{−iθ_{21}(η_{0})} is actually obtained by analytic continuation along η_{0} of the quantity yielding the geometric phase when considered on the real axis. When H(t) is real symmetric, Im θ_{21}(η_{0}) = 0 and we recover the Landau-Zener-Dykhne formula. However, in general, Im θ_{21}(η_{0}) ≠ 0, thus giving rise to a non-trivial prefactor. This prefactor was independently discovered by Berry [3]. The (non-trivial) questions of existence of dissipative paths and competition between several crossing points in S are analyzed in details in [12]. In case no dissipative path above z_{0} exists, an exponential bound follows from integration along dissipative paths close to the real axis [12], [20]. See also Theorem 2. We refer the reader to tables 1 and 2 in [16] and references therein for similar asymptotic formulae in more general 2-level systems as well as in non-generic situations.

When H(t) is an n×n matrix with n > 2 the same strategy is applicable in principle. Let us see this for n = 3, to fix the ideas, in the following simple setting. Suppose there exist two distinct crossing points z_{0} and z_{1} only in S, with Im z_{j} > 0. Let z_{0} be a generic crossing point for the analytic continuations of e_{1}(t) and e_{2}(t) around a loop η_{0} encircling z_{0} only, as above. Assume the analytic continuation of e_{3}(t) in a neighborhood of η_{0} is analytic at z_{0}, as is generically the case. By hypothesis, z_{1} is a generic crossing point for the analytic continuations of e_{2}(t) and e_{3}(t) around a similar loop η_{1} encircling z_{1} only, whereas the analytic continuation of e_{1}(t) in a neighbourhood of η_{1} is analytic at z_{1}. We get, as above, that (4) holds and must be completed by

c_{2}(z|η_{1}) = e^{iθ_{32}(η_{1})}e^{i ∫_{η_{1}} e_{2}(z′)dz′/ε}c_{3}(z), (6)

where θ_{32}(η_{1}) is defined similarly. Assume the composition of the two negatively oriented loops, based at the origin, η_{0} followed by η_{1} can be deformed into one negatively oriented loop, based at the origin, encircling both z_{0} and z_{1}. Then, if c̃_{1} denotes the analytic continuation of c_{1} along a path going from −∞ to +∞ above z_{0} and z_{1}, we get from (4) and (6) c_{3}(+∞) = e^{−iθ_{21}(η_{0})}e^{−i ∫_{η_{0}} e_{1}(z′)dz′/ε}e^{−iθ_{32}(η_{1})}e^{−i ∫_{η_{1}} e_{2}(z′)dz′/ε}c̃_{1}(+∞). Again, it remains to prove c̃_{1}(+∞) = 1 + O(ε) with initial conditions c_{j}(−∞) = δ_{1j}. Such an estimate can be proven in a dissipative domain, defined as follows. A domain P ⊂ S is dissipative if it extends from −∞ to +∞ and if any point z ∈ P can be reached from −∞ by a dissipative path β_{2} ⊂ P for the indices {1, 2} and by a dissipative path β_{3} ⊂ P for the indices {1, 3}. However, generally, dissipative domains going above z_{0} and z_{1} do not extend from −∞ to +∞ [6], [9]. The way out is to perform a perturbative analysis.

Assume H(t, δ) depends on a supplementary parameter δ ≥ 0, satisfies H1 for each fixed δ and is regular enough in (t, δ). Suppose that for δ = 0, the (analytic) eigenvalues e_{j}(t, δ = 0), t ∈ R, labelled in increasing order at t = −∞, are non degenerate, except for two generic real crossings at t_{0} < t_{1} where e_{1}(t_{0}, 0) = e_{2}(t_{0}, 0) and e_{1}(t_{1}, 0) = e_{3}(t_{1}, 0). Assume these degeneracies are lifted as soon as δ > 0, turning t_{0} and t_{1} into avoided crossings. As a consequence, for δ > 0 small enough, there exist crossing points z_{0}(δ) and z_{1}(δ), with the properties discussed above and lim_{δ→0} z_{j}(δ) = t_{j}. Furthermore, the existence of a suitable dissipative domain P can be proven perturbatively so that we get Theorem 1’ [11] Assume H(t, δ) is a 3 × 3 matrix satisfying the hypotheses (loosely) stated in the previous paragraph. Then, for δ, ε > 0 small enough,

P_{31}(ε) = e^{2Im θ_{21}(η_{0},δ)}e^{2Im ∫_{η_{0}} e_{1}(z′,δ)dz′/ε}e^{2Im θ_{32}(η_{1},δ)}e^{2Im ∫_{η_{1}} e_{2}(z′,δ)dz′/ε}(1 + O(ε)), (7)

where O(ε) is uniform in δ and lim_{δ→0} Im θ_{ij}(η_{k}, δ) = lim_{δ→0} Im ∫_{η_{k}} e_{j}(z′, δ)dz′ = 0. Remarks: The above result and generalizations in [11], [18] show that it is possible to get asymptotic formulae for certain transition probabilities in n-level systems displaying avoided crossings, despite the difficulties induced by the notion of dissipativity. The same approach was used when two levels only display one avoided crossing [10]. It yields back Theorem 1 without need to check dissipativity conditions in the complex plane. Moreover, assuming the generic behaviour e_{2}(t) − e_{1}(t) ≃^{δ→0} \sqrt{a^{2}t^{2} + δ^{2}}, we get the Landau-Zener formula: 2 Im ∫_{η_{0}} e_{1}(z′, δ)dz′/ε = −\frac{πδ^{2}}{2aε}(1+O(δ)) and Im θ_{21}(η_{0}, δ) = O(δ) [10]. Further refinements of the Landau-Zener formula can be found in [27].

3 Exponential estimates

Let us come back to general abstract time-dependent Hamiltonians. In such cases we can prove that the transition probability across the gap is exponentially small. Theorem 2 [13] Let H(t) be a time-dependent Hamiltonian satisfying assumptions H1 and H2. Then, there exist C, γ > 0 and ε_{0} > 0 such that for all 0 < ε < ε_{0},

P(+∞, −∞, ε) = lim_{{t_{0}→−∞ \atop t_{1}→+∞}} ∥P_{2}(t_{1})U(t_{1}, t_{0})P_{1}(t_{0})∥^{2} ≤ Ce^{−2γ/ε} (8)

The proof is made by integrating (1) in the complex plane along generalized dissipative paths suitable for operators (see [13], section 5), following [12]. A weaker result can be found in [19]. Similar estimates were known for finite dimensional systems of ODE’s, see e.g. [7] and the literature quoted in [6]. Theorem 2 was then recovered by different methods, some of which allowing a better control on the exponential decay rate γ as a function of the gap g. See [26] for an approach using microlocal analysis, [30] and [15] for superadiabatic techniques, see below, and [33] for a pseudo-differential point of view. It follows from [15] and [26] that γ ≥ cg, g large, for some constant c.

4 Superadiabatic renormalization and reduction theory The root of this method is the work [8] and subsequent generalizations and adapta-

tions [29], [31], [14] on iterative schemes. Since the work [21], the Adiabatic Theorem is often proven by showing that the adiabatic evolution V (t, s) defined by the equation

iε\frac{∂}{∂t}V (t, s) = (H(t) + iε[P′_{1}(t), P_{1}(t)])U(t, s) , V (s, s) = I , ∀t, s ∈ R (9)

satisfies the intertwining property V (t, s)P_{j}(s) = P_{j}(t)V (t, s) and ∥U(t, s) − V (t, s)∥ = O(ε) [1]. Such an approach can be used in a wider context, see [23]. An iterative scheme consists in generating from H(t), by standard perturbation in ε, a sequence of Hamiltonians {H_{q}(t, ε)}_{q∈N} with the following features:

i) The Hamiltonians H_{q}(t, ε) share the same general properties as H(t). In particular, the gap hypothesis is true for H_{q}(t, ε) so that the spectral projectors P_{j,q}(t, ε) are well defined and tend to P_{j}(t) as ε → 0. ii) The adiabatic evolution V_{q}(t, s), defined by (9) with H_{q}(t, ε) and P_{j,q}(t, ε) in place of H(t) and P_{j}(t), approximates U(t, s) up to order O(ε^{q}) instead of O(ε).

It follows that the transition probability between spectral subspaces of H_{q}(t, ε) is O(ε^{2q}) instead of O(ε^{2}) for finite times. Berry [2] formally showed for two-level systems that one can push the estimates up to exponential order by truncating the scheme at optimal q ∈ N. In the present setting, Nenciu proved such exponential estimates in [30] using closely related ideas. The result of Nenciu was recovered in [15] by optimal truncation of the iterative scheme proposed in [14]. Theorem 3 [30], [15] Assume H1 and H2. Then there exist constants C, γ, ε_{0} > 0 and a self adjoint operator H_{∗}(t, ε) defined on D such that for all 0 < ε < ε_{0} and t ∈ R, ∥H_{∗}(t, ε)−H(t)∥ ≤ Cε/(1+|t|)^{(1+α)}. Moreover, the adiabatic evolution V_{∗}(t, s) associated with H_{∗}(t, ε) satisfies sup_{t,s∈R} ∥V_{∗}(t, s) − U(t, s)∥ ≤ Ce^{−γ/ε} as well as V_{∗}(t, s)P_{j,∗}(s, ε) = P_{j,∗}(t, ε)V_{∗}(t, s). Finally, γ ≥ Cg, for g large. Remarks: Theorem 2 becomes a corollary of Theorem 3, since P_{j,∗}(t, ε) →^{t→±∞} P_{j}(±∞). The evolution V_{∗}(t, s) is called superadiabatic evolution due to the exponential estimate.

The main interest of this construction, however, is that it allows to set up a rigorous reduction theory. Assume P_{1}(t)H is finite dimensional, say two-dimensional, such that σ_{1}(t) = {e_{1}(t), e_{2}(t)} where e_{j}(t) satisfy the gap assumption. Then the transition probability P_{21}(ε) between eigenspaces associated with e_{1}(−∞) and e_{2}(+∞) can be computed modulo errors O(e^{−γ/ε}) by replacing the evolution U(t, s) by the superadiabatic evolution V_{∗}(t, s) which describes the evolution inside the P_{j,∗}(t, ε)H. This leads to an effective two-dimensional problem, with a corresponding effective 2 × 2 Hamiltonian shown to be a perturbation of H(t)P_{1}(t) and to have the same analyticity properties as H(t) in S. Hence, provided the conditions on the analytic continuations of e_{j} stated in Theorem 1 are satisfied, we get with the same notations [15]

P_{21}(ε) = e^{2Imθ_{21}(η_{0})}e^{2Im ∫_{η_{0}} e_{1}(z)dz/ε}(1 + O(ε)) + O(e^{−γ/ε}) (10)

The condition γ > |2Im ∫_{η_{0}} e_{1}(z)dz| is satisfied if g is large enough [15] or in the avoided crossing regime described in Theorem 1’ [10], [11].

Another use of the superadiabatic renormalization consists in performing the analysis of section 2 for finite dimensional systems when the decomposition (2) is replaced by ψ(t) = ∑_{j=1}^{n} c_{j,∗}(t)e^{−i ∫_{0}^{t} e_{j,∗}(t′,ε)dt′/ε}φ_{j,∗}(t, ε) where e_{j,∗}(t, ε) and φ_{j,∗}(t, ε) are the eigenvalues and eigenvectors of H_{∗}(t, ε). After showing that they have analytic continuations in S with similar properties as those of e_{j} and φ_{j}, we get very accurate asymptotic formulae by making use of Theorem 3 ”in the complex plane”. Under the conditions of Theorem 1

P_{21}(ε) = e^{2Imθ_{21,∗}(ε)}e^{2Im ∫_{η_{0}} e_{1,∗}(z,ε)dz/ε}(1 + O(e^{−γ/ε}))), γ > 0, (11) as shown in [17], [11]. The loop η_{0} and θ_{21,∗}(ε) are defined as in section 2.

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