A Nonlinear Landau-Zener Formula

Rémi Carles; Clotilde Fermanian-Kammerer

doi:10.1007/s10955-013-0785-x

Journal of Statistical Physics

August 2013, Volume 152, Issue 4, pp 619–656 | Cite as

A Nonlinear Landau-Zener Formula

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Rémi Carles
Clotilde Fermanian-KammererEmail author

Article

First Online: 11 July 2013

Received: 05 December 2012

Accepted: 12 June 2013

147 Downloads
1 Citations

Abstract

We consider a system of two coupled ordinary differential equations which appears as an envelope equation in Bose–Einstein Condensation. This system can be viewed as a nonlinear extension of the celebrated model introduced by Landau and Zener. We show how the nonlinear system may appear from different physical models. We focus our attention on the large time behavior of the solution. We show the existence of a nonlinear scattering operator, which is reminiscent of long range scattering for the nonlinear Schrödinger equation, and which can be compared with its linear counterpart.

Keywords

Nonlinear scattering Landau-Zener formula Eigenvalue crossing

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Appendix: Rigorous Derivation for a Double-Well Potential

A.1 Mathematical Framework

We give more details concerning the derivation of (1.5) in the case of a condensate in a double well. This is achieved by adapting the approach from [27], from which several intermediary results are borrowed (see also [1] for some refinements to [27] in the confining, one-dimensional case). We shall derive the model (A.3) as an envelope equation in the semi-classical limit. Rewrite (2.3) in the presence of physical constants:

$$ i\hbar\frac{{\partial}\psi^\hbar }{{\partial}t} +\frac{\hbar ^2}{2}\Delta \psi^\hbar= V^\hbar(t,x) \psi^\hbar+ \epsilon( \hbar)\bigl|\psi^\hbar\bigr|^2\psi^\hbar, $$

(A.1)

where ϵ(ħ) is a coupling constant whose value will be discussed later on. Note that we consider a slightly more general framework than in Sect. 2: $x\in\mathbb{R}^{d}$, with d≥1. We assume that V ^ħ(t,x)=V _s(x)+κ(ħ)tV _a(x), with V _s and V _a independent of ħ.

We first describe the assumptions performed on the potential V _s and discuss the first consequences.

Assumption A.1

(Symmetric potential)

The potential $V_{s}\in{\mathcal{C}}^{\infty}(\mathbb{R}^{d})$ is a smooth real-valued function such that:

(1)

The potential V _s is at most quadratic,
$$ {\partial}^\alpha V_s\in L^\infty\bigl( \mathbb{R}^d\bigr),\quad\forall|\alpha|\geq2. $$
(2)

V _s is symmetric with respect to the first coordinate:
$$ V_s(-x_1,x_2,\dots,x_d)= V_s(x_1,x_2,\dots,x_d),\quad \forall x\in\mathbb{R}^d. $$
(3)

V _s admits two minima at x=x _±, where x ₋ and x ₊ are distinct and symmetric with respect to the first axis. Moreover,
$$ V_s(x)>V_s^{\mathrm{min}}= V(x_\pm),\quad \forall x\in\mathbb{R}^d,\ x\neq x_{\pm}, $$
and
$$ V_s^{\mathrm{min}} < \liminf_{|x|\to\infty} V_s(x)=:V_\infty^-. $$
(4)

The minima x ₊ and x ₋ are non-degenerate critical points: ∇V(x _±)=0 and ∇² V(x _±)>0.

Remark A.2

As noted in [27], the last assumption (non-degeneracy) is not crucial.

We denote by

$$ H_0 = -\frac{\hbar^2}{2}\Delta+V_s. $$

The operator H ₀ admits a self-adjoint realization (still denoted by H ₀) on $L^{2}(\mathbb{R}^{d})$ (see e.g. [26]). Let σ(H ₀)=σ _d∪σ _ess be the spectrum of the self-adjoint operator H ₀, where σ _d denotes the discrete spectrum and σ _ess denotes the essential spectrum. It follows that

$$ {\sigma}_d \subset\bigl(V_s^{\mathrm{min}},V_\infty^- \bigr)\quad\text{and}\quad{\sigma}_{\mathrm{ess}}=\bigl[V_\infty^-,+ \infty\bigr). $$

Furthermore, the following two lemmas hold, which follow from [2]:

Lemma A.3

(Lemma 1 from [27])

For any ħ∈(0,ħ ^∗], for some ħ ^∗ fixed, it follows that:

(i)

σ _d is not empty and, in particular, it contains two eigenvalues at least.
(ii)

The lowest two eigenvalues $\lambda_{\pm}^{\hbar}$ of H ₀ are non-degenerate, in particular, $\lambda_{+}^{\hbar}<\lambda_{-}^{\hbar}$. There exists C>0, independent of ħ, such that
$$ \lambda_\pm^\hbar= V_s^{\mathrm{min}}+ \mathcal{O}(\hbar);\quad\inf_{\lambda\in{\sigma}(H_0)\setminus [\lambda_+^\hbar,\lambda_-^\hbar]} \bigl(\lambda- \lambda_\pm^\hbar\bigr)\geq C\hbar. $$

Lemma A.4

([2], and Lemma 2 from [27])

Let $\varphi_{\pm}^{\hbar}$ be the normalized eigenvectors associated to $\lambda_{\pm}^{\hbar}$, then:

(i)

$\varphi_{\pm}^{\hbar}$ can be chosen to be real-valued functions such that
$$ \varphi_\pm^\hbar(-x_1,x_2,\dots, x_d) = \pm\varphi_\pm^\hbar (x_1,x_2,\dots, x_d). $$
(ii)

$\varphi_{\pm}^{\hbar}\in\varSigma\cap L^{\infty}(\mathbb {R}^{d})$, where
$$ \varSigma=\bigl\{f\in H^1\bigl(\mathbb{R}^d\bigr),\ x \mapsto|x| f(x)\in L^2\bigl(\mathbb{R}^d\bigr)\bigr\}. $$
(iii)

There exists C independent of ħ such that for all ħ∈(0,ħ],
$$\begin{aligned} & \bigl\|\varphi_\pm^\hbar\bigr\|_{L^p(\mathbb{R}^d)}\leq C \hbar^{-\frac{d}{2} (\frac{1}{2}-\frac{1}{p} )},\quad\forall p\in [2,\infty], \\ & \bigl\|\nabla\varphi_\pm^\hbar\bigr\|_{L^2(\mathbb{R}^d)}\leq C\hbar ^{-1/2},\quad\bigl\|x \varphi_\pm^\hbar \bigr\|_{L^2(\mathbb{R}^d)}\leq C. \end{aligned}$$

The single-well states are then defined as in (2.5). They satisfy:

$\varphi_{R}^{\hbar}(-x_{1},x_{2},\dots,x_{d})= \varphi_{L}^{\hbar}(x_{1},x_{2},\dots,x_{d})$.
$\|\varphi_{L}^{\hbar}\varphi_{R}^{\hbar}\|_{L^{\infty}} = \mathcal {O} ( \mathrm{e}^{-c/\hbar} )$ for all c<Γ, where Γ denotes the Agmon distance between the two wells
$$ \varGamma=\inf_{\gamma\text{ connecting the two wells}} \int _\gamma\sqrt {V_s(x)- V_s^{\mathrm{min}}}dx>0. $$
For any r>0, there exists C>0 such that
$$ \int_{B(x_+,r)}\bigl|\varphi_R^\hbar(x)\bigr|^2dx =1 +\mathcal{O} \bigl(\mathrm{e}^{-C/\hbar} \bigr),\qquad\int _{B(x_-,r)}\bigl|\varphi_L^\hbar(x)\bigr|^2dx =1 +\mathcal{O} \bigl(\mathrm{e}^{-C/\hbar} \bigr). $$

We denote by

$$ \varPi_c^\hbar= 1- \bigl( \bigl\langle \varphi_+^\hbar,\cdot\bigr\rangle\varphi_+^\hbar+ \bigl \langle\varphi_-^\hbar,\cdot\bigr\rangle\varphi_-^\hbar \bigr) $$

the projection onto the eigenspace orthogonal to the bi-dimensional space associated to $\lambda_{\pm}^{\hbar}$. Finally, we define the splitting between the lowest two eigenvalues

$$ \omega^\hbar= \frac{\lambda_-^\hbar-\lambda _+^\hbar}{2}. $$

(A.2)

Then for all c<Γ, $\omega^{\hbar}=\mathcal{O} ( \mathrm {e}^{-c/\hbar} )$.

We now describe the assumptions made on the potential V _a.

Assumption A.5

(Antisymmetric potential)

The potential $V_{a}\in{\mathcal{C}}^{\infty}(\mathbb{R}^{d})$ is a smooth real-valued function such that:

(1)

The potential V _a is bounded, as well as its derivatives,
$$ {\partial}^\alpha V_a\in L^\infty\bigl( \mathbb{R}^d\bigr),\quad\forall|\alpha|\geq0. $$
(2)

V _a is antisymmetric with respect to the first coordinate:
$$ V_a(-x_1,x_2,\dots,x_d)= -V_a(x_1,x_2,\dots,x_d),\quad \forall x\in\mathbb{R}^d. $$

A.2 An Approximation Result

In this section, we prove:

Proposition A.6

Let d≤2. Let V ^ħ be as in Sect. A.1, with

$$ \kappa=\kappa(\hbar) = \eta\frac{(\omega^\hbar)^2}{\hbar}, $$

for some $\eta\in\mathbb{R}$ independent of ħ. Suppose that ϵ(ħ) is given by

$$ \epsilon(\hbar) = \delta\omega^\hbar\hbar^{d/2}, $$

where δ≥0 does not depend on ħ. Suppose also that the initial datum is of the form

$$ \psi^\hbar(0,x) = \alpha_L\varphi_L^\hbar(x)+ \alpha_R\varphi_R^\hbar(x),\quad \alpha_L,\alpha_R\in\mathbb{C}\text{ \textit{independent of} } \hbar. $$

Define the approximation solution ψ _app by

$$ \psi_{\mathrm{app}}^\hbar(t,x) =a_L \biggl( \frac{\omega^\hbar t}{\hbar} \biggr) \varphi_L^\hbar(x)+a_R \biggl(\frac{\omega^\hbar t}{\hbar} \biggr)\varphi_R^\hbar(x), $$

where (a _L,a _R)=(a _L(τ),a _R(τ)) solves

$$ \left\{ \begin{aligned}[c] i{\partial}_\tau a_L& = \eta\tau a_L - a_R + \delta^\hbar|a_L|^2 a_L;\quad a_{L\mid\tau=0}=\alpha_L, \\ i{\partial}_\tau a_R& = -\eta\tau a_R - a_L+ \delta^\hbar|a_R|^2 a_R;\quad a_{R\mid\tau=0}=\alpha_R, \end{aligned} \right. $$

(A.3)

and

$$ \delta^\hbar= \delta\hbar^{d/2}\int_{\mathbb{R}^d} \varphi_L^4= \delta\hbar^{d/2}\int _{\mathbb{R}^d}\varphi_R^4 $$

is uniformly bounded in ħ∈(0,ħ ^∗]. Then we have the following error estimate: there exist c,C independent of ħ such that for all γ<Γ,

$$ \sup_{| t|\leq c \sqrt{\hbar}/\omega^\hbar}\bigl\|\psi^\hbar(t)- \mathrm{e}^{-it(\lambda _-^\hbar+\lambda_+^\hbar)/(2\hbar)} \psi_{\mathrm{app}}^\hbar(t)\bigr\|_{L^2}\leq C \mathrm{e}^{-\gamma /\hbar}. $$

Remark A.7

The case d=1 and δ<0 could be considered as well, leading to the same result. Considering this case would just make the proof a bit longer, and we refer to [27] for the adaptation.

Remark A.8

Since the range for the slow time τ=ω ^ħ t/ħ is $-c/\sqrt{\hbar}\leq\tau\leq c/\sqrt{\hbar}$, the above approximation result is a large time result, which is consistent with the large time study of (A.3), or, more generally, of (1.5) to which one reduces thanks to the change of variables $s=\sqrt{\eta}\tau$ (we then have G=−η ^−1/2 and δF(u)=η ^−1/2 δ ^ħdiag(|u ₁|²,|u ₂|²)). When ħ is small, Theorem 1.3 gives asymptotics for the profiles a _L and a _R of $\psi^{\hbar}_{\mathrm{app}}$ for large times ($|t|\leq c{\sqrt{\hbar}/ \omega^{\hbar}}$) and the connexion between the profiles for t<0 and t>0 involves the Landau-Zener transition coefficient $\mathrm{ e}^{-{\pi\over\eta^{2}}}$ (at leading order when ħ goes to 0).

Proof

For simplicity, we write the proof for t≥0 and it naturally extends to t≤0.

Step 1: Preliminaries. We begin by proving estimates on ψ ^ħ, then we perform the rescaling suggested by the form of the approximation solution. First, it follows from Lemma A.4 and [4] that for fixed ħ>0, (A.1) has a unique, global solution $\psi^{\hbar}\in C(\mathbb{R};\varSigma)$ (Σ is defined in Lemma A.4). In addition, if we set

$$\begin{aligned} \text{Mass: } &M^\hbar(t) = \bigl\|\psi^\hbar(t) \bigr\|_{L^2}^2, \\ \text{Energy: }& E^\hbar(t) = \frac{1}{2}\bigl\|\hbar\nabla \psi^\hbar(t)\bigr\|_{L^2}^2 + \frac{\epsilon(\hbar)}{2}\bigl\| \psi^\hbar(t)\bigr\|_{L^4}^4 + \int _{\mathbb{R}^d}V(t,x) \bigl|\psi^\hbar(t,x)\bigr|^2dx , \end{aligned}$$

then we have the a priori estimates:

$$\begin{aligned} &\frac{d M^\hbar}{dt} =0, \\ & \frac{d E^\hbar}{dt} = \int_{\mathbb{R}^d} {\partial}_t V(t,x) \bigl|\psi^\hbar(t,x)\bigr|^2dx= \kappa(\hbar)\int _{R^d} V_a(x)\bigl|\psi^\hbar (t,x)\bigr|^2dx , \end{aligned}$$

where we have used (2.4) for the last equality. Note that (A.1) is not a Hamiltonian equation, unlike the one studied in [1], where the Hamiltonian structure is crucial. The conservation of mass and the form of the initial data yield

$$ M^{\hbar}(t)= \bigl\|\psi^\hbar(t)\bigr\|_{L^2} = \bigl\| \psi^\hbar(0)\bigr\|_{L^2} =M^\hbar(0)\leq C, $$

for some C independent of ħ. We have

$$\begin{aligned} E^\hbar(0)&= \bigl\langle\alpha_L\varphi_L^\hbar+ \alpha_R\varphi_R^\hbar, \alpha_L H_0 \varphi_L^\hbar+ \alpha_R H_0\varphi_R^\hbar \bigr\rangle+ \frac{\epsilon(\hbar)}{2}\bigl\|\alpha_L\varphi _L^\hbar+ \alpha_R\varphi_R^\hbar \bigr\|_{L^4}^4 . \end{aligned}$$

Introduce $\varOmega^{\hbar}=(\lambda_{-}^{\hbar}+\lambda_{+}^{\hbar})/2$ and notice that $\varOmega^{\hbar}=V_{s}^{\mathrm{min}} +\mathcal{O}(\hbar )$ by Lemma A.3. Noting the identities

$$ H_0\varphi_L^\hbar= \varOmega^\hbar\varphi_L^\hbar- \omega^\hbar\varphi_R^\hbar;\qquad H_0\varphi_R^\hbar= \varOmega^\hbar \varphi_R^\hbar- \omega^\hbar \varphi_L^\hbar, $$

(A.4)

we infer from Lemma A.3 and Lemma A.4 that

$$ E^\hbar(0)= V_s^{\mathrm{min}} M^\hbar(0) + \mathcal{O}(\hbar). $$

Therefore, since V _a is bounded,

$$\begin{aligned} \bigl\|\hbar\nabla\psi^\hbar(t)\bigr\|_{L^2}^2 & \leq2 E^\hbar(t) -2 V_s^{\mathrm{min}} M^\hbar(t)+C \kappa(\hbar)tM^\hbar(t) \\ &\leq2 E^\hbar(0) -2 V_s^{\mathrm{min}} M^\hbar(0)+2\kappa(\hbar)\int_0^t \int_{R^d} V_a(x)\bigl|\psi^\hbar(t,x)\bigr|^2dx+C \kappa(\hbar)t \\ &\leq C\hbar+ C \frac{(\omega^\hbar)^2}{\hbar} t. \end{aligned}$$

Next, as in [27], we consider the slow time

$$ \tau= \frac{\omega^\hbar t}{\hbar}\geq0, $$

and the new unknown function

$$ \varPsi^\hbar(\tau,x) = \psi^\hbar(t,x) \mathrm{e}^{i\varOmega^\hbar t/\hbar} = \psi^\hbar(t,x) \mathrm{e}^{i\varOmega^\hbar\tau /\omega^\hbar} . $$

It solves

$$ i{\partial}_\tau\varPsi^\hbar= \frac{1}{\omega^\hbar} \bigl(H_0-\varOmega^\hbar\bigr) \varPsi^\hbar+\eta\tau V_a \varPsi^\hbar+\delta \hbar^{d/2}\bigl|\varPsi^\hbar\bigr|^2\varPsi^\hbar, $$

(A.5)

and in view of the above estimates,

$$ \bigl\|\varPsi^\hbar(\tau)\bigr\|_{L^2} = \bigl\| \varPsi^\hbar(0)\bigr\|_{L^2}=\mathcal{O}(1);\quad\bigl\|\nabla \varPsi^\hbar(\tau)\bigr\|_{L^2}^2 \leq C \biggl( \frac{1}{\hbar} + \frac{\omega^\hbar}{\hbar^2}\tau\biggr). $$

(A.6)

We decompose Ψ ^ħ as

$$ \varPsi^\hbar= \varphi^\hbar+ \psi_c^\hbar,\quad\psi_c^\hbar= \varPi_c^\hbar\varPsi^\hbar, $$

(A.7)

so we can write φ ^ħ as

$$ \varphi^\hbar(\tau,x) = {\mathtt{a}}^\hbar_L( \tau)\varphi_L^\hbar(x) + {\mathtt{a}}^\hbar _R(\tau)\varphi_R^\hbar(x) , $$

for some complex-valued coefficients ${\mathtt{a}}^{\hbar}_{L}$ and ${\mathtt{a}}^{\hbar}_{R}$. Projecting (A.5), and using (A.4), we find:

$$\begin{aligned} i\dot{\mathtt{a}}^\hbar_L & = -{\mathtt{a}}^\hbar _R +\eta\tau\int_{\mathbb{R}^d}V_a \varPsi ^\hbar\varphi_L^\hbar+ \delta \hbar^{d/2} \int_{\mathbb{R}^d} \bigl|\varPsi^\hbar\bigr|^2 \varPsi^\hbar\varphi_L^\hbar, \\ i\dot{\mathtt{a}}^\hbar_R & = -{\mathtt{a}}^\hbar _L +\eta\tau\int_{\mathbb{R}^d}V_a \varPsi ^\hbar\varphi_R^\hbar+ \delta \hbar^{d/2} \int_{\mathbb{R}^d} \bigl|\varPsi^\hbar\bigr|^2 \varPsi^\hbar\varphi_R^\hbar, \\ i{\partial}_\tau\psi_c^\hbar& = \frac{1}{\omega^\hbar} \bigl(H_0-\varOmega^\hbar\bigr) \psi_c^\hbar+\eta\tau\varPi^\hbar_c \bigl(V_a \varPsi^\hbar\bigr)+\delta \hbar^{d/2}\varPi_c^\hbar\bigl(\bigl| \varPsi^\hbar\bigr|^2 \varPsi^\hbar\bigr). \end{aligned}$$

The proof of Proposition A.6 consists in showing that ${\mathtt{a}}^{\hbar}_{L/R}$ are close to a _L/R, and that $\psi ^{\hbar}_{c}$ is small. This is achieved in two more steps.

Step 2. A priori estimates on $\mathtt {a}^{\hbar}_{R}$, ${\mathtt{a}}^{\hbar}_{L}$ and $\psi^{\hbar}_{c}$ . By definition, we have ${\mathtt{a}}^{\hbar}_{L}=\int_{\mathbb {R}^{d}}\varPsi ^{\hbar}\varphi _{L}^{\hbar}$, so Cauchy–Schwarz inequality yields

$$ \bigl\|\varphi^\hbar(\tau)\bigr\|^2_{L^2}= \bigl|{ \mathtt{a}}^\hbar_L(\tau)\bigr|^2+ \bigl|{ \mathtt{a}}^\hbar_R(\tau)\bigr|^2\leq \bigl(M^\hbar\bigr)^2 \bigl(\bigl\|\varphi_L^\hbar \bigr\|_{L^2}^2 + \bigl\|\varphi_R^\hbar \bigr\|_{L^2}^2 \bigr)\leq2 \bigl(M^\hbar \bigr)^2. $$

Decompose the nonlinearity acting on Ψ ^ħ as

$$ \bigl|\varPsi^\hbar\bigr|^2 \varPsi^\hbar= \bigl| \varphi^\hbar\bigr|^2\varphi^\hbar+ \mathtt{R}^\hbar. $$

Using the a priori estimates on a ^ħ and Lemma A.4, we have

$$ \bigl\|\bigl|\varphi^\hbar\bigr|^2\varphi^\hbar\bigr \|_{L^2}\leq\bigl\|\varphi^\hbar\bigr\|_{L^\infty}^2 \bigl\|\varphi^\hbar\bigr\|_{L^2}\leq C \bigl(\bigl\|\varphi _L^\hbar \bigr\|_{L^\infty}^2 + \bigl\|\varphi^\hbar_R\bigr\| _{L^\infty}^2 \bigr)\leq C\hbar^{-d/2}. $$

Since we have the pointwise estimate

$$ \bigl|\mathtt{R}^\hbar\bigr|\leq C \bigl( \bigl|\varphi^\hbar \bigr|^2 \bigl|\psi^\hbar_c\bigr| + \bigl|\psi^\hbar _c\bigr|^3 \bigr), $$

we infer

$$ \bigl\|\mathtt{R}^\hbar\bigr\|_{L^2}\leq C \bigl( \hbar^{-d/2} \bigl\|\psi^\hbar_c\bigr\|_{L^2} + \bigl\|\psi^\hbar_c\bigr\|^3_{L^6} \bigr), $$

and Gagliardo–Nirenberg inequality yields (d≤2)

$$ \bigl\|\psi^\hbar_c\bigr\|^3_{L^6(\mathbb{R}^d)} \leq C \bigl\| \psi^\hbar_c\bigr\|_{L^2(\mathbb{R}^d)}^{3-d}\bigl\|\nabla \psi^\hbar_c\bigr\|_{L^2(\mathbb{R}^d)}^{d}. $$

Since d≤2, we can factor out $\|\psi^{\hbar}_{c}\|_{L^{2}(\mathbb {R}^{d})}$, and (A.6) gives

$$ \bigl\|\psi^\hbar_c\bigr\|^3_{L^6(\mathbb{R}^d)} \leq C \bigl\| \psi^\hbar_c\bigr\|_{L^2(\mathbb{R}^d)}\hbar^{-d/2} \biggl(1 + \biggl(\frac {\omega^\hbar \tau}{\hbar} \biggr)^{d/2} \biggr). $$

Therefore,

$$ \bigl\|\mathtt{R}^\hbar\bigr\|_{L^2}\leq C \hbar^{-d/2} \biggl(1 + \biggl(\frac{\omega^\hbar \tau}{\hbar} \biggr)^{d/2} \biggr)\bigl\|\psi^\hbar_c\bigr\|_{L^2}, $$

(A.8)

and |Ψ ^ħ|² Ψ ^ħ satisfies a similar estimate. We infer

$$ \bigl|\dot{\mathtt{a}}^\hbar_L(\tau) \bigr|+ \bigl| \dot{\mathtt{a}}^\hbar_R(\tau)\bigr|\leq C (1+\tau) +C \biggl(1 + \biggl(\frac{\omega^\hbar\tau}{\hbar} \biggr)^{d/2} \biggr) . $$

Since d≤2 and $\omega^{\hbar}=\mathcal{O}(\mathrm{e}^{-c/\hbar })$, we can simplify the above estimate:

$$ \bigl|\dot{\mathtt{a}}^\hbar_L(\tau)\bigr|+ \bigl| \dot{\mathtt{a}}^\hbar_R(\tau) \bigr|\leq C (1+\tau), \quad\forall\tau\geq0. $$

From this we infer

$$ \bigl\|{\partial}_\tau\varphi^\hbar \bigr\|_{L^2} \leq\bigl|\dot{\mathtt{a}}^\hbar_L( \tau) \bigr|\bigl\|\varphi_L^\hbar\bigr\|_{L^2} + \bigl| \dot{\mathtt{a}}^\hbar_R(\tau) \bigr|\bigl\| \varphi_R^\hbar\bigr\|_{L^2} \leq C(1+\tau), $$

(A.9)

and, again from Lemma A.4,

$$ \bigl\|{\partial}_\tau\bigl(\bigl| \varphi^\hbar\bigr|^2\varphi^\hbar\bigr) \bigr\|_{L^2} \leq3\bigl\|\varphi^\hbar\bigr\|_{L^\infty}^2 \bigl\|{\partial}_\tau\varphi^\hbar\bigr\|_{L^2} \leq C\hbar^{-d/2}(1+\tau). $$

(A.10)

Step 3. Stability estimates. In view of (A.7), we first prove that $\psi^{\hbar}_{c}$ is small. Since $\psi^{\hbar}_{c\mid\tau=0}=0$, Duhamel’s formula yields

$$\begin{aligned} \psi^\hbar_c(\tau) &= -i\eta\int _0^\tau\mathrm{e}^{-i(H_0-\varOmega^\hbar)(\tau-s)/\omega^\hbar} \bigl(s \varPi^\hbar_c \bigl( V_a \varPsi^\hbar\bigr) (s) \bigr)ds \\ &\quad-i \delta\hbar^{d/2}\int_0^\tau \mathrm{e}^{-i(H_0-\varOmega^\hbar)(\tau-s)/\omega^\hbar} \varPi ^\hbar_c \bigl( \bigl| \varPsi^\hbar\bigr|^2 \varPsi^\hbar\bigr) (s)ds. \end{aligned}$$

(A.11)

Each term is treated in a similar fashion: when Ψ ^ħ is replaced by φ ^ħ, we perform an integration by parts, and for the remaining term, we use directly the a priori estimates. For the first part of (A.11), we write

$$ \varPi^\hbar_c \bigl( V_a \varPsi^\hbar\bigr)= \varPi^\hbar_c \bigl( V_a \varphi^\hbar\bigr) + \varPi^\hbar_c \bigl( V_a \psi^\hbar_c \bigr), $$

and set

$$\begin{aligned} I^\hbar_1(\tau) & = \int_0^\tau \mathrm{e}^{-i(H_0-\varOmega^\hbar)(\tau-s)/\omega^\hbar} \bigl (s \varPi^\hbar_c \bigl( V_a \varphi^\hbar\bigr) (s) \bigr)ds, \\ I^\hbar_2(\tau) & = \int_0^\tau \mathrm{e}^{-i(H_0-\varOmega^\hbar)(\tau-s)/\omega^\hbar} \bigl (s \varPi^\hbar_c \bigl( V_a \psi^\hbar_c \bigr) (s) \bigr)ds. \end{aligned}$$

Integrating by parts,

$$\begin{aligned} I^\hbar_1(\tau) & = -i\omega^\hbar \mathrm{e}^{-i(H_0-\varOmega^\hbar)(\tau-s)/\omega^\hbar} \bigl (H_0-\varOmega^\hbar \bigr)^{-1}\varPi^\hbar_c \bigl(s V_a \varphi^\hbar\bigr) (s) \bigr|_0^\tau \\ &\quad+i\omega^\hbar\int_0^\tau \mathrm{e}^{-i(H_0-\varOmega^\hbar)(\tau-s)/\omega^\hbar} \bigl (H_0-\varOmega^\hbar \bigr)^{-1} \bigl(\varPi^\hbar_c \bigl( V_a \varphi^\hbar\bigr)+ s \varPi^\hbar_c \bigl( V_a {\partial}_\tau\varphi^\hbar\bigr) \bigr) (s)ds . \end{aligned}$$

From Lemma A.3, there exists C independent of ħ∈(0,ħ ^∗] such that

$$ \bigl\|\hbar\bigl(H_0-\varOmega^\hbar \bigr)^{-1}\varPi^\hbar_c \bigr\| _{L^2\to L^2}\leq C. $$

We infer, using (A.9),

$$ \bigl|I^\hbar_1(\tau) \bigr|\leq C\frac{\omega^\hbar}{\hbar} \bigl(\tau+ \tau^3 \bigr). $$

For $I^{\hbar}_{2}$, we have directly

$$ \bigl|I^\hbar_2(\tau) \bigr|\leq C\int_0^\tau s\bigl\|\psi_c^\hbar(s)\bigr\|_{L^2}ds. $$

For the nonlinear term in Duhamel’s formula (the second term of (A.11)), we also write

$$\varPi^\hbar_c\bigl(\bigl|\varPsi^\hbar\bigr|^2 \varPsi^\hbar\bigr)= \varPi^\hbar_c\bigl(\bigl| \varphi^\hbar\bigr|^2\varphi^\hbar\bigr)+ \varPi^\hbar_c\mathtt{R}^\hbar, $$

and set

$$\begin{aligned} I^\hbar_3(\tau) & =\hbar^{d/2} \int _0^\tau\mathrm{e}^{-i(H_0-\varOmega^\hbar)(\tau-s)/\omega^\hbar} \varPi^\hbar_c \bigl( \bigl|\varphi^\hbar\bigr|^2 \varphi^\hbar\bigr) (s)ds, \\ I^\hbar_4(\tau) & = \hbar^{d/2}\int _0^\tau\mathrm{e}^{-i(H_0-\varOmega^\hbar)(\tau-s)/\omega^\hbar} \varPi^\hbar_c \mathtt{R}^\hbar(s)ds. \end{aligned}$$

We have, directly from (A.8),

$$ \bigl|I^\hbar_4(\tau)\bigr|\leq C\int_0^\tau \biggl(1 + \biggl(\frac{\omega^\hbar s}{\hbar} \biggr)^{d/2} \biggr)\bigl\| \psi_c^\hbar(s)\bigr\|_{L^2}ds, $$

and performing an integration by parts for $I_{3}^{\hbar}$, using (A.10), we have

$$ \bigl|I^\hbar_3(\tau) \bigr|\leq C\frac{\omega^\hbar}{\hbar} \bigl(1+ \tau^2 \bigr). $$

Since d≤2 and ω ^ħ decays exponentially, we come up with:

$$ \bigl\|\psi_c^\hbar(\tau)\bigr\|_{L^2}\leq C \biggl( \frac{\omega^\hbar }{\hbar} \bigl(1+\tau^3 \bigr) + \int_0^\tau(1+s) \bigl\|\psi_c^\hbar(s)\bigr\| _{L^2}ds \biggr). $$

Gronwall lemma yields

$$ \bigl\|\psi_c^\hbar(\tau)\bigr\|_{L^2} \leq C \frac{\omega^\hbar}{\hbar}\bigl(1+\tau^3\bigr)\mathrm {e}^{C(\tau+\tau^2)}. $$

Recalling again that ω ^ħ decays exponentially, we can write that for all c ₀<Γ (the Agmon distance between the two wells),

$$ \bigl\|\psi_c^\hbar(\tau)\bigr\|_{L^2} \leq C \bigl(1+ \tau^3\bigr)\mathrm{e}^{C(\tau+\tau^2)-c_0/\hbar}. $$

This is small as ħ→0, provided that τ ²≪1/ħ. More precisely, there exist c ₁,c ₂>0 independent of ħ such that

$$ \bigl\|\psi_c^\hbar(\tau)\bigr\|_{L^2} \leq C \mathrm{e}^{-c_1/\hbar},\quad0\leq\tau\leq\frac {c_2}{\sqrt{\hbar}}. $$

(A.12)

To conclude the proof of the proposition, set

$$ \mathtt{w}^\hbar_L={\mathtt{a}}^\hbar _L-a_L;\qquad\mathtt{w}^\hbar _R= {\mathtt{a}}^\hbar_R-a_R. $$

Subtracting the equation for a _L from the equation for $\mathtt {a}^{\hbar}_{L}$, we have

$$\begin{aligned} i\dot{\mathtt{w}}_L^\hbar&= -\mathtt{w}^\hbar _R +\eta\tau\int_{\mathbb{R}^d}V_a \bigl( \varPsi^\hbar- a_L\varphi^\hbar_L \bigr)\varphi_L^\hbar \\ &\quad+\delta\hbar^{d/2}\int_{\mathbb{R}^d} \bigl(\bigl| \varPsi^\hbar\bigr|^2\varPsi^\hbar -|a_L|^2a_L \bigl|\varphi_L^\hbar\bigr|^2 \varphi_L^\hbar\bigr)\varphi_L^\hbar. \end{aligned}$$

We have

$$ \int_{\mathbb{R}^d}V_a \bigl(\varPsi^\hbar- a_L\varphi^\hbar_L \bigr) \varphi_L^\hbar= \int_{\mathbb{R}^d}V_a \bigl({\mathtt{a}}^\hbar_R \varphi^\hbar_R + \psi_c^\hbar+\mathtt{w}^\hbar _L \varphi^\hbar_L \bigr) \varphi_L^\hbar, $$

therefore, since the product $\varphi_{L}^{\hbar}\varphi_{R}^{\hbar}$ decays exponentially in ħ,

$$ \biggl|\int_{\mathbb{R}^d}V_a \bigl( \varPsi^\hbar- a_L\varphi^\hbar_L \bigr)\varphi_L^\hbar\biggr|\leq C \bigl( \mathrm{e}^{-c/\hbar} + \bigl\|\psi^\hbar_c \bigr\|_{L^2} + \bigl|\mathtt{w}^\hbar_L\bigr| \bigr). $$

A similar estimate can be established for the other source term in the equation for $\mathtt{w}^{\hbar}_{L}$. The equation for $\mathtt {w}^{\hbar}_{R}$ is handled in the same fashion, and using (A.12), we end up with:

$$ \bigl|\dot{\mathtt{w}}_L^\hbar(\tau)\bigr| + \bigl|\dot{ \mathtt{w}}_R^\hbar(\tau)\bigr|\leq C \bigl( \bigl| \mathtt{w}^\hbar_L(\tau)\bigr| +\bigl|\mathtt{w}^\hbar _R(\tau)\bigr| + \mathrm{e}^{-c/\hbar} \bigr), \quad0\leq\tau \leq \frac{c_2}{\sqrt{\hbar}}. $$

Gronwall lemma and (A.12) then yield Proposition A.6. □

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Rémi Carles
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Clotilde Fermanian-Kammerer
- 2
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1.CNRS & Univ. Montpellier 2MontpellierFrance
2.LAMA, UMR CNRS 8050Université Paris ESTCréteil CedexFrance

A Nonlinear Landau-Zener Formula

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