QUANTUM ADIABATIC EVOLUTION^{a}

Alain Joye^{b}and Charles-Edouard Pfister^{c}

bCentre de Physique Théorique

CNRS Marseille Luminy Case 907 F-13288 Marseille France cEcole Polytechnique Fédérale de Lausanne Département de Mathématiques CH-1015 Lausanne Switzerland

ADIABATIC THEOREM OF QUANTUM MECHANICS

The notion of adiabatic evolution or adiabatic process is an important theoretical concept, which occurs at several places in Physics. The main feature of this concept is that although the process is very slow, global changes can take place without local changes. Adiabaticity is at the border between dynamics and statics. This concept occurred in Classical Mechanics first and was introduced by Boltzmann. The notion of adiabatic invariant plays an important role here. In Statistical Mechanics or Thermodynamics we say that a process is adiabatic if the state of the system at time t is a Gibbs state for the Hamiltonian H(t). Similarly in Quantum Mechanics, 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) remains isolated during the time-interval [t_{0}, t_{1}]. Even if H(t_{0}) = H(t_{1}), the eigenfunction ψ(t_{1}) is generally different form ψ(t_{0}) by a phase which can be decomposed into a dynamical phase related to the energy-level e(t) and a geometric phase related to the spectral subspaces visited during the adiabatic process. This is the fundamental observation of Berry ^{1} which gave rise to extensive developments during the last ten years ^{2},^{3},^{4}. We also mention that in the Old Quantum Mechanics the Adiabatic Principle formulated mainly by Ehrenfest played a crucial role in the quantization of observables.

The Adiabatic Theorem in Quantum Mechanics has been proven very early by Born and Fock ^{5}. It was further refined at several occasions, in particular the papers of Kato ^{6} and Garrido ^{7} brought new ideas into the subject. We also mention works by Lenard ^{8} and Nenciu ^{9}. In this short paper we describe recent mathematical results obtained during the last four years, mainly in the case of analytic time-dependent quantum systems, where almost no rigorous results were known before. Let H(t) be the Hamiltonian of a time-dependent system. We suppose that H(t) is a self-adjoint operator, uniformly bounded from below, defined on a dense domain D, independent of t, of a Hilbert space H. We suppose that H(t) is at least of class C^{2}. The time evolution of the system is governed by the evolution operator U(s, s′), solution of the Schrödinger equation (h̄ = 1)

aLecture delivered by C.-E.Pfister

i\frac{∂}{∂s}U(s, s′) = H(εs)U(s, s′), U(s′, s′) = I I (1)

where 1/ε is the typical time-scale over which the Hamiltonian changes significantly. This equation is conveniently studied using a rescaled time, t = εs, which leads to

iε\frac{∂}{∂t}U(t, t′) = H(t)U(t, t′), U(t′, t′) = I I (2)

where, with an abuse of notations, we have again denoted the evolution by U. It should be stressed, although we do not write it explicitly, that U(t, t′) is a function of ε. The adiabatic limit corresponds to the limit ε tending to zero which is a singular limit. The Adiabatic Theorem states that there exists an approximate solution V (t, t′) of equation (2), called Adiabatic Evolution, such that

sup_{t∈[t_{0},t_{1}]} ∥U(t, t_{0}) − V (t, t_{0})∥ = O(ε) (3)

The main assumptions needed to prove this result are the smoothness of the Hamiltonian and the Gap Condition: for each t the spectrum σ(t) of H(t) can be separated into two closed subsets σ_{1}(t), σ_{2}(t) so that these two sets of IR^{2} satisfy

dist(σ_{1}(t), σ_{2}(t)) ≥ g

for some positive constant g. By convention σ_{1}(t) is always a bounded subset of σ(t). We denote the spectral projector associated with σ_{1}(t) by Q(t). The main feature of the Adiabatic Evolution V (t, t′) is that it is ”compatible with the decomposition of the Hilbert space H” into

H = Q(t)H ⊕ (I I − Q(t))H (4)

This is expressed by the intertwining property

V (t, t′)Q(t′) = Q(t)V (t, t′) (5)

A more explicit expression of V (t, t′) can be given when σ(t) is discrete, see next section.

Let us consider now the consequences of (3) and (5). Let ψ_{0} be an initial wavefunction at t = t_{0}, such that Q(t_{0})ψ_{0} = ψ_{0}. At time t the wave-function is ψ(t) = U(t, t_{0})ψ_{0}, and ϕ(t) = V (t, t_{0})ψ_{0} is an approximate solution. Because of property (5)

Q(t)ϕ(t) = ϕ(t) (6)

i.e. ϕ(t) belongs to the spectral subspace Q(t)H of H(t). In the special case where dimQ(t) ≡ 1, then ψ_{0} is an eigenfunction of H(t_{0}) for the eigenvalue e(t_{0}) = trQ(t_{0})H(t_{0}) and ϕ(t) an eigenfunction of H(t) for the eigenvalue e(t) = trQ(t)H(t). The information (6), combined with (3), implies that the probability that the system makes a transition from the subspace Q(t_{0})H to the subspace (Q(t)H)^{⊥} is of order ε^{2}.

It is not very difficult to show, that V (t, t′) is solution of the equation (see e.g. ^{10})

iε\frac{∂}{∂t}V (t, t′) = (H(t) + iε [\frac{d}{dt}Q(t), Q(t)]) V (t, t′), V (t′, t′) = I I (7)

This result can be improved in the following way, provided H(t) is smooth enough. If H(t) is of class C^{q+1}, q ≥ 1, one can find through an iterative scheme based on regular perturbation theory another decomposition of the Hilbert space H into

H = Q_{k}(t)H ⊕ (I I − Q_{k}(t))H (8)

where k = 0, 1, 2, ⋯, q−1 and Q_{0}(t) = Q(t). The projector Q_{k}(t) is a spectral projector of the Hamiltonian H_{k}(t) := H(t) − iε [\frac{d}{dt}Q_{k−1}(t), Q_{k−1}(t)], with H_{0}(t) = H(t). The solution V_{k} of

iε\frac{∂}{∂t}V_{k}(t, t′) = (H(t) + iε [\frac{d}{dt}Q_{k}(t), Q_{k}(t)]) V_{q−1}(t, t′), V_{k}(t′, t′) = I I (9)

is then compatible with the decomposition (8), and

sup_{t∈[t_{0},t_{1}]} ∥U(t, t_{0}) − V_{k}(t, t_{0})∥ = O (ε^{k+1}) (10)

V_{k} is called Adiabatic Evolution of order k and V_{0} = V. In order to define the operators V_{k}(t, t′) we must find the projections Q_{k}(t). They are constructed by the iterative scheme mentioned above ^{11},^{12}, as approximate solutions of the Heisenberg equation

iε\frac{d}{dt}A(t) = [H(t), A(t)] (11)

If it happens that all derivatives of the resolvent of H(t) are zero up to order p, then Q_{j}(t) = Q(t), j ≤ p. In particular, if this happens at times t_{0} and t_{1}, then the wave function ψ(t_{1}) = U(t_{1}, t_{0})ψ_{0} can be approximated by the function ϕ(t_{1}) = V_{p}(t_{1}, t_{0})ψ_{0}, such that Q(t_{1})ϕ(t_{1}) = ϕ(t_{1}) and ∥ψ(t_{1}) − ϕ(t_{1})∥ = O (ε^{p}). Let us summarize this discussion and give further improvements in the analytic case in the next paragraph.

Adiabatic Theorem

A) Let H(t) be of class C^{q+1}, q ≥ 1, and let the Gap Condition be satisfied for all t ∈ [t_{0}, t_{1}]. Then there exists ε_{k} > 0 so that for all ε < ε_{k}, the Adiabatic Evolution of order k, V_{k}, 0 ≤ k ≤ q − 1 is defined in [t_{0}, t_{1}], and

sup_{t∈[t_{0},t_{1}]} ∥U(t, t_{0}) − V_{k}(t, t_{0})∥ = O (ε^{k+1})

B) Let H(t) have an analytic continuation in the strip S_{α} = {t + is ∈ I C : |s| < α}. Let the Gap Condition be satisfied for all t ∈ IR, and let c(t) be an integrable function, with lim_{t→±∞} c(t) = 0, and H^{+}, H^{−} be two self-adjoint operators such that for all φ ∈ D

sup_{|s|<α} ∥(H(t + is) − H^{+})φ∥ ≤ c(t)(∥φ∥ + ∥H^{+}φ∥)

for t >> 1 and

sup_{|s|<α} ∥(H(t + is) − H^{−})φ∥ ≤ c(t)(∥φ∥ + ∥H^{−}φ∥)

for t << 1.

Then there exist ε^{∗} > 0 and a positive constant κ, such that for all ε < ε^{∗} and for all t, t′ ∈ IR we have a decomposition of H into

H = Q_{∗}(t)H ⊕ (I I − Q_{∗}(t))H

and a Superadiabatic Evolution V_{∗}(t, t′) satisfying,

∥U(t, t′) − V_{∗}(t, t′)∥ = O (e^{−κ/ε})

and

∥Q(t) − Q_{∗}(t)∥ = O(ε)

Moreover lim_{t→±∞} ∥Q(t) − Q_{∗}(t)∥ = 0.

Comments:

1) In case B), since the function c(t) is integrable, there exists a constant K such that ∥U(t, t′) − V_{∗}(t, t′)∥ ≤ Ke^{−κ/ε} for any t and t′. Therefore if ψ(t) is a normalized solution of the Schrödinger equation such that lim_{t→−∞} ∥(I I − Q(t))ψ(t)∥ = 0, then the transition probability to the spectral subspace (I I − Q(+∞))H at time t = +∞ is exponentially small

lim_{t→+∞} ∥(I I − Q(t))ψ(t)∥^{2} = O (e^{−2κ/ε}) (12)

This result was first obtained in ^{13} in the case of matrices, and in the general case in ^{14}. The method of the proof is quite different than the above mentioned method. The evolution is studied along special complex paths in the complex time-plane. The method gives good estimations of κ in the case of matrices and even optimal values of κ in the case of 2 × 2 matrices. Recently Martinez ^{15} has rederived result (12) by a microlocal analysis. We also mention ^{16}, however the results are weaker.

2) For a proof of the above theorem, based on the iteration scheme see ^{12}. Nenciu^{17} gave a closely related proof, which precedes ^{12}. Recently Sjöstrand ^{18} obtained similar results by a microlocal analysis.

3) The evolution V_{∗} is called a Superadiabatic Evolution because of the exponential estimate. It is obtained as follows. Using the analyticity condition we can prove that there exists a positive constant c, such that for all q ≤ q^{∗}(ε) = [1/ecε], and

∥U(t, t′) − V_{q}(t, t′)∥ = O (c^{q}ε^{q}q!) (13)

Choosing q = q^{∗}(ε) we get the exponential bound. Thus V_{∗} is an adiabatic evolution of order O(1/ε).

TWO-LEVEL SYSTEMS

We consider here only the case of analytic systems with two energy levels e_{1}(t) and e_{2}(t), which have the same permanent degeneracy. As above Q(t) is the spectral projection associated with e_{1}(t). We suppose that trH(t) ≡ 0, so that H(t)^{2} = ρ(t)I I with ρ(t) ≥ g^{2}/4 in order that the Gap Condition be satisfied. We have

e_{1}(t) = −\sqrt{ρ(t)}, \sqrt{1} = 1 (14)

and

Q(t) = \frac{1}{2} (I I + \frac{H(t)}{e_{1}(t)}) (15)

We suppose that the conditions of part B) of the Adiabatic Theorem are satisfied. For these simple systems the Adiabatic Evolution can be written more explicitly as

V (t, t′) = exp (−i/ε ∫_{t′}^{t} e_{1}(s)ds) W (t, t′)Q(t′)
+ exp (−i/ε ∫_{t′}^{t} e_{2}(s)ds) W (t, t′)(I I − Q(t′))  (16)

where W (t, t′) is solution of

i\frac{∂}{∂t}W (t, t′) = i [\frac{d}{dt}Q(t), Q(t)] W (t, t′), W (t′, t′) = I I (17)

W (t, t′) has the intertwining property

W (t, t′)Q(t′) = Q(t)W (t, t′) (18)

Expression (16) gives the decomposition of V into a dynamical part, the exponential factors, and an evolution W of geometrical nature, which can be interpreted as describing a parallel transport.

Let χ_{−} ∈ Q(−∞)H and χ_{+} ∈ (I I − Q(+∞))H be two normalized eigenvectors of H^{−} and H^{+}. The transition probability from χ_{−} to χ_{+} is given by

P(ε, χ_{−}, χ_{+}) = lim_{{t_{0}→−∞ \atop t_{1}→+∞}} |⟨χ_{+}|U(t_{1}, t_{0})χ_{−}⟩|^{2} (19)

Let us suppose that the Hamiltonian H(t) is analytic in a strip S_{α}. In order of derive an asymptotic formula for P(ε, χ_{−}, χ_{+}), we observe that the evolution U(t, t_{0}) is analytic in S_{α} whereas V (t, t_{0}) is multivalued and has singularities at the zeros of ρ. Indeed, let us assume that there exists a complex eigenvalue crossing at z_{j} (and by symmetry at \overline{z_{j}}), i.e. ρ(z_{j}) = 0. Clearly, from the expressions (14) and (15) the eigenvalues and eigenprojections have a singularity at z_{j} and \overline{z_{j}}. We further suppose that z_{j} is a zero of order one for ρ(z) with Imz_{j} > 0. We can continue analytically e_{i}(t), i = 1, 2, and Q(t) in the punctured strip S_{α}, which is defined by removing the zeros of ρ from S_{α},but the analytic continuation is not unique. We fix a point t of the real axis and make the analytic continuation of e_{1}(t) and Q(t) along a clockwise oriented simple loop γ_{j} based at t, encircling z_{j} and no other zero of ρ. After analytic continuation we obtain the values e_{1}(t|γ_{j}) = e_{2}(t), and Q(t|γ_{j}) = (I I − Q(t)). A simple computation shows that

i [\frac{d}{dt}Q(t), Q(t)] = \frac{i}{4ρ(t)} [\frac{d}{dt}H(t), H(t)] (20)

and therefore the generator of W is a single-valued analytic function in the punctured strip. W itself is multi-valued and W (t|γ_{j}), the analytic continuation of W (t, t) along γ_{j}, is such that

W (t|γ_{j})Q(t) = (I I − Q(t))W (t|γ_{j}) (21)

It should be noticed that W (t|γ_{j}) is not unitary anymore since γ_{j} is a path in the complex plane. W (t|γ_{j}) describes the transport of vectors along γ_{j}. If we start with a vector in Q(t), then, at the end of the loop γ_{j}, we get a vector in the orthogonal subspace. When the eigenvalues are not degenerate, we can thus define a complex phase θ_{j}(t) by the relation

W (t|γ_{j})ϕ_{1}(t) = exp(−iθ_{j}(t))ϕ_{2}(t) (22)

where ϕ_{i}(t) is a (normalized) eigenvector of H(t) for the eigenvalue e_{i}(t), i = 1, 2. The important point is that W (t|γ_{j}) and θ_{j}(t) depend only on the homotopy class of the closed path γ_{j} based at t in the punctured strip. Moreover, Imθ_{j}(t) is independent of the base point t of γ_{j} since t is real.

It is natural to write the evolution U(t, 0) as U(t, 0) := V (t, 0)A(t). Using the fact that the evolution U is analytic in the strip, i.e. path independent, whereas V is path dependent, we can derive a useful identity for A by considering the Schrödinger equation along two different paths, the real axis and a path η_{j} in the complex plane going above z_{j} ^{19}. This identity allows us to express the transition probability (19) in terms of the analytic continuations of the eigenvalues and eigenprojections along the path γ_{j} and of A(·) along the path η_{j} ^{19}. We can perform the asymptotic analysis of the transition probability (19) as ε → 0 whenever we have a good control of the analytic continuation of A along η_{j}. At his step it is necessary to impose further conditions on the behaviour of the multi-valued function Φ(z) := −2 ∫_{t}^{z} \sqrt{ρ}(z′)dz′. These conditions specify the global behaviour in the strip of the so-called Stokes lines, which are the level-lines ImΦ(·) = Φ(z_{j}), where z_{j} is a zero of ρ ^{13},^{19}. The results of the analysis are given in table 1 when the above mentioned conditions are satisfied for one (and then necessarily unique) simple zero z_{j} of ρ. This case is generic. In the table we have dropped the index j.

Table 1. Asymptotic transition probability in the generic case.

Characteristics of H(t)  Transition probability P(ε, χ_{−}, χ_{+})

General formula^{19}

Complex hermitian dimQ(t)H > 1

\begin{aligned}e & \frac{2}{ε} Im ∫_{γ} e_{1}(z)dz & ( & |⟨χ_{+}|W (+∞, t)W (t|γ)W (t, −∞)χ_{−}⟩| & 2 \\ & & +O(ε)), ε → 0 \end{aligned}

Dykhne formula^{20} Real symmetric

dimQ(t)H = 1 e^{\frac{2}{ε} Im ∫_{γ} e_{1}(z)dz} (1 + O(ε)) , ε → 0

Geometrical prefactor^{21},^{13},^{22} Complex hermitian

dimQ(t)H = 1 e^{\frac{2}{ε} Im ∫_{γ} e_{1}(z)dz}e^{2Imθ} (1 + O(ε)) , ε → 0

Landau-Zener formula^{23},^{24} Complex hermitian dimQ(t)H = 1 Avoided crossing:

e_{2}(t) − e_{1}(t) ≃^{δ→0}

\sqrt{a^{2}t^{2} + 2ctδ + b^{2}δ^{2}}

\begin{matrix}e^{− \frac{πδ^{2}}{2ε} (\frac{b^{2}}{a} − \frac{c^{2}}{a^{3}} )(1+O(δ))} (1 + O(δ) + O(ε)), \\ ε, δ → 0 \end{matrix}

Accurate formula^{25} Complex hermitian dimQ(t)H = 1

\begin{matrix}e^{\frac{2}{ε} Im ∫_{γ} e_{1}^{∗}(z,ε)dz}e^{2Imθ^{∗}(ε)} (1 + O (e^{−κ/ε})), \\ ε → 0, κ > 0 \end{matrix}

Remarks:

Notice that we can express the decay-rate as

2 Im ∫_{γ_{j}} e_{1}(z)dz = 2Im ∫_{t}^{z_{j}}(e_{1}(z) − e_{2}(z))dz = Φ(z_{j}) < 0 (23)

with the integration-path from t to z inside γ_{j}. Except for the real case and the Landau-Zener formula, we have a nontrivial prefactor of geometric nature which is

0 ≤ |⟨χ_{+}|W (+∞, t)W (t|γ)W (t, −∞)χ_{−}⟩| ≤ ∥W (t|γ)∥ (24) and which reduces to exp (2Imθ) when dimQ(t)H = 1. This geometric prefactor was measured recently in a spin experiment^{22}. The last row of the table yields an asymptotic formula accurate up to exponentially small relative errors. The expressions e_{1}^{∗} and θ^{∗} are defined as e_{1} and θ with H_{q^{∗}} in place of H (see comment 3 on the Adiabatic Theorem).

In Table 2 we have given the results in the non generic case, when there is an eigenvalue crossing, which is a zero of ρ(z) of higher order and/or when several eigenvalue crossings (say q) determine the asymptotics of the transition probability. These cases give rise to different prefactors and/or interference phenomena in the transition probability. The above outlined method must be slightly modified.

Table 2. Asymptotic transition probability in non generic cases.

Characteristics of H(t)  Transition probability P(ε, χ_{−}, χ_{+})

High order degeneracies^{26},^{27} Real symmetric dimQ(t)H = 1 ρ(z) ≃ (z − z_{1})^{n} H(z) ≃ (z − z_{1})^{m}

\begin{matrix}e^{\frac{2}{ε} Im ∫_{γ} e_{1}(z)dz} (4 sin^{2} (\frac{π(n−2m)}{2(n+2)}) + O (ε^{p})) , \\ p > 0, ε → 0 \end{matrix}

Stückelberg oscillations^{28},^{25} Complex hermitian dimQ(t)H = 1 z_{1}, z_{2}, two simple zeros of ρ(z) contribute

\begin{matrix}|∑_{j=1}^{q} e^{− \frac{i}{ε} ∫_{γ_{j}} e_{1}(z)dz}e^{−iθ_{j}} |^{2} + O(ε)e^{− \frac{2}{ε} |Im ∫_{γ_{1}} e_{1}(z)dz|}, \\ ε → 0 \end{matrix}

Conjugated effects^{26} Real symmetric dimQ(t)H = 1 z_{1}, z_{2}, characterized by (n_{1}, m_{1}), (n_{2}, m_{2}) as above contribute

\begin{matrix}|∑_{j=1}^{q} 2σ_{j} sin (\frac{π(n_{j}−2m_{j})}{2(n_{j}+2)}) e^{− \frac{i}{ε} ∫_{γ_{j}} e_{1}(z)dz}|^{2} + \\ O(ε^{p})e^{− \frac{2}{ε} |Im ∫_{γ_{1}} e_{1}(z)dz|}, σ_{j} = ±1, ε → 0 \end{matrix}

Accurate formula^{25} Complex hermitian dimQ(t)H = 1 z_{1}, z_{2}, two simple zeros of ρ(z) contribute

\begin{matrix}|∑_{j=1}^{q} e^{− \frac{i}{ε} ∫_{γ_{j}} e_{1}^{∗}(z)dz}e^{−iθ_{j}^{∗}} |^{2} + \\ O (e^{−κ/ε}) e^{− \frac{2}{ε} |Im ∫_{γ_{1}} e_{1}(z)dz|}, κ > 0 ε → 0 \end{matrix}

For the precise definition of σ_{j} see ^{26}. Notice that 2m_{j} < n_{j}, and the value of

Im ∫_{γ_{j}} e_{1}(z)dz < 0 (25)

and is the same for all j = 1, …, q. In the last line the energy e_{1}^{∗}, or the phases θ_{j}^{∗} are defined using H_{q^{∗}}, and we have

Im ∫_{γ_{j}} e_{1}^{∗}(z)dz = Im ∫_{γ_{j}} e_{1}(z)dz + O(ε^{2}) (26)

In the discussion only the eigenvalue-crossings in the upper half-plane play a role, because we consider the transition from the lower energy-level to the higher energy-level.

Finally we mention that these methods, the iteration scheme and the analysis in the complex plane, can be applied to study similar problems, like the semi-classical limit of the above barrier reflection coefficient for the stationary Schrödinger equation, or the adiabatic invariant of a time-dependent classical oscillator ^{25}.

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