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JOINT EXCITATION PROBABILITY FOR TWO HARMONIC
OSCILLATORS IN DIMENSION ONE AND THE MOTT PROBLEM
GIANFAUSTO DELL’ANTONIO, RODOLFO FIGARI, AND ALESSANDRO TETA
Abstract. We analyze a one dimensional quantum system consisting of a test particle inter- acting with two harmonic oscillators placed at the positions a 1 , a 2 , with a 1 > 0, |a 2 | > a 1 , in the two possible situations: a 2 > 0 and a 2 < 0. At time zero the harmonic oscillators are in their ground state and the test particle is in a superposition state of two wave packets centered in the origin with opposite mean momentum. Under suitable assumptions on the physical pa- rameters of the model, we consider the time evolution of the wave function and we compute the probability P n− 1 n 2 (t) (resp. P n+ 1 n 2 (t)) that both oscillators are in the excited states labelled by n 1 , n 2 > 0 at time t > |a 2 |v− 0 1 when a 2 < 0 (resp. a 2 > 0). We prove that P n− 1 n 2 (t) is negligible with respect to P n+ 1 n 2 (t), up to second order in time dependent perturbation theory. The system we consider is a simplified, one dimensional version of the original model of a cloud chamber introduced by Mott in [M], where the result was argued using euristic arguments in the framework of the time independent perturbation theory for the stationary Schr¨odinger equation. The method of the proof is entirely elementary and it is essentially based on a stationary phase argument. We also remark that all the computations refer to the Schr¨odinger equation for the three-particle system, with no reference to the wave packet collapse postulate.
- Introduction, notation and result
In his paper of 1929 Mott ([M]) analyzes the dynamics of formation of tracks left an α-particle emitted by a radioactive source inside the supersaturated vapour in a cloud chamber. He notices the difficulty to understand intuitively how a spherical wave function, describing the particle isotropically emitted by the source, might manifest itself as a straight track in the cloud chamber. Without referring to any wave packet collapse, he proposes an explanation based on the analysis of the whole quantum system made up of the α-particle and of the atoms of the vapour. Using a simplified model with only two atoms and making use of time independent perturbation arguments, he concludes that each ionization process focuses the probability of presence of the α-particle on narrower and narrower cones, around the straight line connecting the source to the ionized atoms. 1
2 GIANFAUSTO DELL’ANTONIO, RODOLFO FIGARI, AND ALESSANDRO TETA
In this way Mott suggests a quantum dynamical mechanism responsible of the transition be- tween an initial superposition of outgoing waves heading isotropically in all directions toward an incoherent (classical) sum of those same waves. We mention that the same problem is also discussed in [H] and later in [Be], where the above approach is compared with the explanation based on the wave packet collapse. We refer to [Br], [HA], [CL], [BPT] for some further elaborations on the subject and to [LR] for a description of the original experimental apparatus. The aim of our work is to provide a detailed time dependent analysis of a one dimensional version of the system investigated by Mott. The system we consider consists of a test particle and two harmonic oscillators. In our model a superposition of two wave packets centered in the origin with opposite momentum plays the role of the spherical wave of the α-particle and the oscillators replace the atoms to be ionized. Under suitable assumptions on the physical parameters of the model, we perform a detailed time analysis of the evolution of the system wave function using time dependent perturbation theory and we give a quantitative estimate of the joint excitation probability of the oscillators. Roughly speaking, our main result is that such probability is essentially zero if the oscillators are placed on opposite sides of the origin, while it has a finite, non-zero value in the other case. Following the line of reasoning of Mott, the result can be interpreted saying that before the interaction the test particle is delocalized while after the interaction it is either on the left (if there is an excited oscillator on the left) or on the right (if there is an excited oscillator on the right). In any case one can say that the test particle propagates along an almost classical trajectory, without making any reference to the wave packet collapse postulate. In [CCF] the authors consider a similar problem in three dimensions where a particle interacts via zero range forces with localized two level quantum systems. A non perturbative analysis of the model is carried out but results are valid only in the scattering regime. Let us introduce the model. We consider a three-particle non relativistic quantum system in dimension one, made of one test particle with mass M interacting with two harmonic oscillators with the identical mass m. We denote by R the position coordinate of the test particle and by r 1 ,r 2 the position coordinates of the two oscillators. The Hamiltonian of the system in L^2 (R^3 ) is
H = H 0 + λH 1 (1.1)
H 0 = −
ℏ^2
2 M
∆R −
ℏ^2
2 m
∆r 1 +
mω^2 (r 1 − a 1 )^2 −
ℏ^2
2 m
∆r 2 +
mω^2 (r 2 − a 2 )^2 (1.2)
H 1 = V (δ−^1 (R − r 1 )) + V (δ−^1 (R − r 2 )) (1.3)
where λ > 0, ω > 0, a 1 > 0, a 2 ∈ R, with a 1 < |a 2 |, δ > 0 and V is a smooth interaction potential. The assumptions on V will guarantee that the Hamiltonian H is self-adjoint with the same domain of H 0 and then the evolution problem corresponding to the Hamiltonian H is well posed. For the test particle we choose an initial state ψ in the form of a superposition
4 GIANFAUSTO DELL’ANTONIO, RODOLFO FIGARI, AND ALESSANDRO TETA
(A 0 )
λ 0 ≡
λ M v^20
(A 1 )
The quantities δm ≡ (^) Mm , δE ≡ (^) M vℏω 2 0 , δR ≡ (^) |aσj | , δL ≡ (^) |aδj | , δτj ≡ (^) ωv|a^0 j | , for j = 1, 2 , are
all O(ε) where
λ 0 ε 1 (1.12)
(A 2 )
The interaction potential V : R → R is a continuous, positive and compactly supported function.
Let us briefly comment on the above assumptions. In (A 0 ) we ensure that the dimensionless coupling constant λ 0 is small. In (A 1 ) we assume that the mass and the kinetic energy of the test particle are much larger than the mass and the spacing of the energy levels of the oscillators; moreover the initial wave packets of the test particle are assumed to be well localized and the interaction is required to be short range; finally the characteristic time of the oscillators ω−^1 is assumed to be much smaller than the flight times τ 1 , τ 2 of the test particle, which are defined by
τ 1 =
a 1 v 0
, τ 2 =
|a 2 | v 0
Condition (1.12) guarantees that the first and second order corrections in perturbation theory remain small compared with the unperturbed wave function, in fact of order λ 0 ε−^1 and λ^20 ε−^2 respectively. In order to understand the meaning of (A 1 ), let us consider the parameters M , v 0 , a 1 a 2 all of order one. Then we obtain m = O(ε), ω = O(ε−^1 ), ℏω = O(ε), σ = O(ε), δ = O(ε). We observe that the length γ introduced in (1.7) can be written as
γ = a 1 δτ 1
δE δm
and this means that γ is of the same order of δ and σ. In particular this guarantees that the transit time of the test particle on the region where the oscillators are localized is of the same order of the characteristic time of the oscillators. To simplify the notation, from now on we shall fix
δ = γ (1.15)
We also introduce here a (large) parameter which is useful to express the various estimates in the proof
5
Λj ≡
|aj | γ
= O(ε−^1 ), j = 1, 2 (1.16)
Our main result is the following.
Theorem 1. Let us assume (A 0 ), (A 1 ), (A 2 ) and fix t > τ 2 , n 1 6 = 0, n 2 6 = 0. Then for any k ∈ N, with k > 2 , we have
P n− 1 n 2 (t) ≤
Λ^21 k−^4
λ 0 ε
C n(k 1 )n 2 (t) (1.17)
P n+ 1 n 2 (t) = 16π^4
π
λ 0 ε
N 2
j=1, 2
V^ ˜(qj )(˜φnj φ 0 )(qj )
2
qj = −nj
δE δm
|Sn 1 n 2 (t)| ≤
λ 0 ε
Dn 1 n 2 (t) (1.20)
where the symbol ˜ denotes Fourier transform and C n(k 1 )n 2 (t), Dn 1 n 2 (t) are functions of the physical parameters of the model which will be explicitely given during the proof (see (4.16), (4.19) below).
We remark that the estimates (1.17), (1.18), (1.20) are not optimal; in particular C( nk 1 )n 2 (t) and
Dn 1 n 2 (t) diverge for t → ∞. From (4.16), (4.19) it will be clear that C n(k 1 )n 2 (t), Dn 1 n 2 (t) are of order one, and then the estimates are meaningful only for t larger but of the same order of τ 2. Let us briefly outline the strategy of the proof and give a heuristic argument which, at least at a qualitative level, justifies the result stated in theorem 1. We find convenient to represent the solution of (1.8), (1.9) in the form
Ψ(R, r 1 , r 2 , t) =
n 1 ,n 2
fn 1 n 2 (R, t)φa n^11 (r 1 )φa n^22 (r 2 ) (1.21)
where fn 1 n 2 (·, t) = fn 1 n 2 (t) belongs to L^2 (R) for any n 1 , n 2 ∈ N and t ≥ 0, and it is explicitely given by
fn 1 n 2 (R, t) =
dr 1 dr 2 φa n^11 (r 1 )φa n^22 (r 2 )Ψ(R, r 1 , r 2 , t) (1.22)
We notice that the coefficients of the expansion fn 1 n 2 (R, t) have a precise physical meaning; in fact the quantity (^) ∫
Ω
dR |fn 1 n 2 (R, t)|^2 (1.23)
7
f (^) n(1) 1 n 2 (t)
= −δn 20
∫ (^) t
0
ds Γn 10 (t − s) V (^) na 110 e−^2 ℏi sE^0 e−^ ℏi sK^0 ψ − δn 10
∫ (^) t
0
ds Γ 0 n 2 (t − s) V (^) na 220 e−^2 ℏi sE^0 e−^
i ℏ sK 0 ψ
≡ δn 20 f (1) n 10 (t) +^ δn 10 f^
(1) n 20 (t)^ (1.34)
From formula (1.34) it is clear that f (^) n(1) 1 n 2 (t) = 0 if n 1 6 = 0 and n 2 6 = 0. As expected, this means that the probability that both oscillators are in an excited state is zero up to first order in perturbation theory. As a consequence, from (1.10) we get
P n± 1 n 2 (t) =
dR
∣f (^) n(2) 1 n 2 (R, t)
∣^2 , n 1 6 = 0, n 2 6 = 0 (1.35)
Following the original strategy of Mott, a crucial point of the analysis is the explicit evaluation
of f (^) n(1) 10 (t) and f (^) n(1) 20 (t). We notice that V (^) na 110 (x) and
e−^ ℏi sK^0 ψ±
(x) are essentially different
from zero only for x ' a 1 and x ' ±v 0 s respectively. This means that the only non zero
contribution to the time integral defining f (^) n(1) 10 (t) comes from ψ+^ and such contribution is
essentially concentrated around s = a v^10 = τ 1. Hence we can argue that f (^) n(1) 10 (t) is approximately given by a wave packet starting at time τ 1 from the position a 1 of the first oscillator, with a velocity close to v 0. In particular it is essentially different from zero only in a neighborhood of a 1 + v 0 (t − τ 1 ), for t > τ 1.
Analogously, f (^) n(1) 20 (t) is approximately given by a wave packet starting at time τ 2 from the position a 2 of the second oscillator, with a velocity close to v 0 if a 2 > 0, and to −v 0 if a 2 < 0.
Then f (^) n(1) 20 (t) is essentially different from zero only in a neighborhood of a 2 + v 0 (t − τ 2 ), for t > τ 2 , a 2 > 0, and in a neighborhood of a 2 − v 0 (t − τ 2 ), for t > τ 2 , a 2 < 0.
Let us now consider the second order term f (^) n(2) 1 n 2 (t); exploiting expression (1.34), we have
f (^) n(2) 1 n 2 (t) = −δn 20
∫ (^) t
0
ds Γn 10 (t − s)
j 1
V (^) na 11 j 1 f (^) j(1) 10 (s) − δn 10
∫ (^) t
0
ds Γ 0 n 2 (t − s)
j 2
V (^) na 22 j 2 f 0 (1)j 2 (s)
∫ (^) t
0
ds Γn 1 n 2 (t − s)V (^) na 110 f 0 (1)n 2 (s) −
∫ (^) t
0
ds Γn 1 n 2 (t − s)V (^) na 220 f (^) n(1) 10 (s) (1.36)
Since we are interested in the probability that both oscillators are excited, only the last two terms of (1.36) are relevant.
We notice that the supports of V (^) na 110 and f (^) n(1) 20 (s) are essentially disjoint for any s ≥ 0 and this implies that the third term in the r.h.s. of (1.36) gives a negligible contribution. For the same reason, the fourth term in the r.h.s. of (1.36) is also approximately zero if a 2 < 0. This explains why we expect that an estimate like (1.17) holds.
8 GIANFAUSTO DELL’ANTONIO, RODOLFO FIGARI, AND ALESSANDRO TETA
On the other hand, in the case a 2 > 0 the product V (^) na 220 f (^) n(1) 10 (s) is different from zero for s ' τ 2. In such case the fourth term in the r.h.s. of (1.36) gives a non zero contribution and this explains why we can expect that a formula like (1.18) holds. We collect here some further notation which will be used in the paper.
- 〈x〉 denotes (1 + x^2 )^1 /^2 ;
- dkxl f is the derivative of order k with respect to xl of a smooth function f (x 1 ,... , xn), for n ∈ N and l = 1,... , n;
- ‖f ‖W (^) sk, 1 =
∑n l=
∑k m=
dx 〈x〉s|(dmxl f )(x)|, k ∈ N, s ≥ 0;
- ‖f ‖L^1 s = ‖f ‖W (^) s^0 ,^1 ;
- c is a generic positive numerical constant. The paper is organized as follows. In section 2 we study the first order approximation step. In section 3 we analyze the second order approximation, distinguishing the two case a 2 > 0 and a 2 < 0. In section 4 we compute the joint excitation probability of the two oscillators concluding the proof of theorem 1. Finally in the appendix we give a proof of a technical lemma.
- First order approximation
In this section we fix t > τj , j = 1, 2, and we give an estimate of the first order terms f (^) n(1)j 0 (t). We only give the details for the case a 2 > 0 since the opposite case can be treated similarly. We rewrite f (^) n(1)j 0 (t) as follows
f (^) n(1)j 0 (t) = f (^) n(1)j 0 , +(t) + f (^) n(1)j 0 , −(t) (2.1)
f (^) n(1)j 0 , ±(t) = −Γnj 0 (t)
∫ (^) t
0
ds einj^ ωs^ e ℏi sK^0 V aj nj 0 e
− (^) ℏi sK (^0) ψ± (^) (2.2)
Moreover let us define for j = 1, 2 and s, t ≥ 0
Ij (s) = e
i ℏ sK^0 V (^) najj 0 e−^ i ℏ sK^0 (2.3)
h± j (t) =
∫ (^) t
0
ds einj^ ωs^ Ij (s)ψ±^ (2.4)
As a first step the operator (2.3) will be written in a more convenient form.
Lemma 2.1. For any f ∈ L^2 (R) and s ≥ 0 the following identity holds
(Ij (s)f )(R) =
dξ gj (ξ)f (R + (M γ)−^1 ℏs ξ) e i (^2) M γℏs 2 ξ^2 ei^
Rγ ξ e−iΛj^ ξ^ (2.5)
10 GIANFAUSTO DELL’ANTONIO, RODOLFO FIGARI, AND ALESSANDRO TETA
where
F (^) j± (R, ξ, s) = gj (ξ) e i (^2) M γℏs 2 ξ^2 ˆ ψ± 1 (R, ξ, s) (2.11)
ψˆ± 1 (R, ξ, s) = √N σ
e−^
(R− Rˆ 1 )^2 2 σ^2 ±i^
Pˆ 1 ± ℏ R^ (2.12)
Rˆ 1 = − ℏ
M γ
ξs, Pˆ 1 ± = P 0 ±
γ
ξ (2.13)
θ± j (ξ, s) =
s τj
ξ − qj
s τj
and qj has been defined in (1.19).
Proof. The proof is trivial if we notice that
ψ±
R + (M γ)−^1 ℏs ξ
ei^
R γ ξ^ = ψˆ± 1 (R, ξ, s) e±iΛj^ ξ^ τs (^1) (2.15)
and use lemma 2.1.
The next step is to estimate (2.10), i.e. an integral containing the rapidly oscillating phase Λj θ± j (ξ, s). The standard stationary (or non-stationary) phase methods can be used to obtain the estimate. It is worth mentioning that the integral in (2.10) contains also other phase factors depending on (ξ, s) which, however, are slowly varying under our assumptions on the physical parameters of the model. The asymptotic analysis for Λj → ∞ is simplified by the fact that θ j± (ξ, s) is a quadratic function. The only critical points of the phase are (±qj , ±τj ) and, moreover, the hessian matrix is non degenerate, with eigenvalues ±τ (^) j− 1. This means that the behaviour of (2.10) for Λj → ∞ in the case with θ− j is radically different from the case with θ+ j , due to the fact that in the first case the critical point never belongs to the domain of integration while in the second case this happens for t > τj. For the analysis of this second case it will be useful the following elementary lemma. For the convenience of the reader a proof of the lemma will be given in the appendix.
Lemma 2.3. Let us consider for any Λ > 0
J (Λ) =
dx
∫ (^) μ
−ν
dy f (x, y) eiΛxy^ (2.16)
where μ, ν are positive parameters, f is a complex-valued, sufficiently smooth function. Then
11
J (Λ) =
K 1 (Λ) (2.17)
2 πf (0, 0) +
Λ^2
K 2 (Λ) (2.18)
2 πf (0, 0) +
Λ^2
2 πi dxdyf (0, 0) +
Λ^3
K 3 (Λ) (2.19)
where Kl(Λ), l = 1, 2 , 3 , are explicitely given (see the appendix) and satisfy the estimates
|K 1 (Λ)| ≤ c 1
‖f (·,0)‖L 1 +
dx‖dxdyf (x,·)‖L 2
|K 2 (Λ)| ≤ c 2
‖d^2 xf (·,0)‖L^1 + ‖dxdyf (·,0)‖L^1 +
dx ‖d^2 xd^2 yf (x,·)‖L^2
|K 3 (Λ)| ≤ c 3
‖d^3 xf (·,0)‖L 1 +‖d^3 xdyf (·,0)‖L 1 +‖d^2 xd^2 yf (·,0)‖L 1 +
dx‖d^3 xd^3 yf (x,·)‖L 2
and the constants c 1 , c 2 , c 3 depend only on μ, ν.
Exploiting lemma 2.3 we obtain the following asymptotic behaviour of (2.10) for t > τj when the phase is θ+ j.
Proposition 2.4. For any t > τj we have
h+ j (t) =
2 πτj Λj
e−iΛj^ qj^ F (^) j+ (·, qj , τj ) +
Λ^2 j
R+ j (·, t, Λj ) (2.23)
where
|R+ j (R, t, Λj )| ≤ Cj
[ ∫
dξ |d^2 ξ F (^) j+ (R, ξ, τj )| +
dξ |dξdsF (^) j+ (R, ξ, τj )|
dξ
(∫ (^) t
0
ds |d^2 ξ d^2 s F (^) j+ (R, ξ, s)|^2
) 1 / 2 ]
and Cj depends on t and τj.
Proof. Let us introduce the change of coordinates x = ξ − qj , y = s − τj in (2.10) and the shorthand notation F (x, y) = e−iΛj^ qj^ F (^) j+ (R, x + qj , y + τj ). Then
13
and integrate by parts two times in the r.h.s. of (2.10) we easily obtain the r.h.s. of (2.27) with
R− j (R, t, Λj ) = −τ (^) j^2
∫ (^) t
0
ds
(s + τj )^2
dξ
d^2 ξ F (^) j− (R, ξ, s)
eiΛj^ θ j− (ξ,s)^ (2.30)
Then by a trivial estimate we conclude the proof.
Collecting together the results of propositions 2.4 and 2.5 we finally obtain an asymptotic expression for t > τj of the first order terms when Λj → ∞
f (^) n(1)j 0 (t) =
A(1) j Λj
e−^ ℏi tK^0 ψ j+ +
Λ^2 j
R(1) j (·, t, Λj ) (2.31)
A(1) j = − 2 πi
λτj ℏ
e −i(nj +1)ωt−iΛj qj +i 2 ℏM γτj 2 q^2 j gj (qj ) (2.32)
ψ+ j = ψˆ 1 + (·, qj , τj ) (2.33) R(1) j (·, t, Λj ) = −Γnj 0 (t)
R− j (·, t, Λj ) + R+ j (·, t, Λj )
We observe that the leading term in the r.h.s. of (2.31) can be more conveniently written in the form
A
(1) j Λj
e−^ ℏi tK^0 ψ+ j = − 2 πi
λ 0 √ δm δE
eiηj^ (t)^ V˜ (qj )˜φnj φ 0 (qj )e−^ ℏi tK^0 ψ+ j (2.35)
ηj (t) =
n^2 j 2
δE δτj
− (nj + 1)ωt +
nj δτj
ψ+ j (R) =
b √ σ
e−^
(R−Rj )^2 2 σ^2 +i^
Pj ℏ R, Rj = nj aj δE, Pj = P 0 (1 − nj δE) (2.37)
Then it is clear that the leading term has the form of a free evolution of a wave packet which starts at t = τj from the position aj of jth^ oscillator, with mean momentum Pj. Notice that under our assumptions Pj ' P 0 > 0. In particular (2.31) gives a precise meaning to the qualitative statement concerning the approx-
imate behavior of f (^) n(1)j 0 (t) made in section 1.
14 GIANFAUSTO DELL’ANTONIO, RODOLFO FIGARI, AND ALESSANDRO TETA
- Second order approximation
In this section we fix t > τ 2 and consider the second order terms corresponding to both oscil- lators in some exited states, i.e. terms of the type (see formula (1.36))
∫ (^) t
0
ds Γnj nl (t − s) V (^) nall 0 f (1),± nj 0 (s)
= i
λ ℏ
Γnlnj (t)
∫ (^) t
0
ds einlωs
∫ (^) s
0
ds′^ einj^ ωs
′ Il(s)Ij (s′)ψ±^ (3.1)
≡ i
λ ℏ
Γnlnj (t) h± jl(t) (3.2)
for j, l = 1, 2 , l 6 = j. Proceeding as in lemmas 2.1 and 2.2, a straightforward computation in the case a 2 > 0 yields
h± jl(t) =
∫ (^) t
0
ds
∫ (^) s
0
ds′
dξ
dη G± jl(·, ξ, s′, η, s) eiΛj^ θ
± j (ξ,s′)+iΛlθ ± l (η,s)^ (3.3)
G± jl(R, ξ, s′, η, s) = gj (ξ)gl(η)ei^
ℏ 2 M γ^2 (s ′ξ (^2) +sη (^2) +2sξη) (^) ˆ ψ 2 ± (R, ξ, s′, η, s) (3.4)
ψˆ± 2 (R, ξ, s′, η, s) = √N σ
e−^
(R− Rˆ 2 )^2 2 σ^2 ±i^
Pˆ 2 ± ℏ R^ (3.5)
Rˆ 2 = − ℏ
M γ
(ξs′^ + ηs), Pˆ 2 ± = P 0 ±
γ
(ξ + η) (3.6)
where gj and θ j± have been defined in (2.6) and (2.14) respectively. In the case a 2 < 0 the same representation formula (3.3) holds if we replace Λ 2 , τ 2 with −Λ 2 , −τ 2 , where Λ 2 = |a 2 |γ−^1 , τ 2 = |a 2 |v− 0 1. In both cases, we shall discuss the asymptotic behaviour of h± jl(t) for Λ 1 , Λ 2 → ∞. The integral (3.3) contains a rapidly oscillating phase and moreover the phase has exactly one critical point. Therefore the behaviour strongly depends on whether or not the critical point lies in the integration domain. We shall analyze separately the two cases a 2 > 0 and a 2 < 0.
3.1. The case a 2 > 0.
We distinguish the four possible cases: i) h+ 21 (t), ii) h− 21 (t), iii) h− 12 (t), iv) h+ 12 (t). It is eas- ily seen that the point (ξ 0 , s′ 0 , η 0 , s 0 ) where the phase is stationary is: (q 2 , τ 2 , q 1 , τ 1 ) for i), (−q 2 , −τ 2 , −q 1 , −τ 1 ) for ii) and (−q 1 , −τ 1 , −q 2 , −τ 2 ) for iii). In all three cases the stationary point of the phase does not belong to the domain of integration and then the integral rapidly decreases to zero for Λ 1 , Λ 2 → ∞. On the other hand in the case iv) the stationary point is (q 1 , τ 1 , q 2 , τ 2 ), i.e. it belongs to the domain of integration and therefore there is a leading term
16 GIANFAUSTO DELL’ANTONIO, RODOLFO FIGARI, AND ALESSANDRO TETA
The r.h.s. of (3.11) can be estimated using (2.18), (2.21) and the result is
|(II)| ≤
C 2
Λ 2 Λ^21
‖d^2 xG(·, 0 , 0 , 0)‖L 1 + ‖dxdyG(·, 0 , 0 ,0)‖L 1 +
dx‖d^2 xd^2 yG(x,·, 0 ,0)‖L 2
C 2
Λ 2 Λ^21
[ ∫
dξ |d^2 ξ G+ 12 (R, ξ, τ 1 , q 2 , τ 2 )| +
dξ |dξds′ G+ 12 (R, ξ, τ 1 , q 2 , τ 2 )|
dξ
(∫ (^) t
0
ds′^ |d^2 ξ d^2 s′ G+ 12 (R, ξ, s′, q 2 , τ 2 )|^2
) 1 / 2 ]
where C 2 is a constant depending on τ 1 , τ 2. The term (III) can be more conveniently written as
(III) =
dw
∫ (^) t−τ 2
−τ 2
dz L(w, z)ei^
Λ 2 τ 2 wz^ (3.13)
L(w, z) ≡
dx
∫ (^) z+τ 2 −τ 1
−τ 1
dy
G(x, y, w, z) − G(x, y, 0 , 0)
ei^
Λ 1 τ 1 xy^ (3.14)
where L(0, 0) = 0. Using (2.19), (2.22) we have
(III) = (IV ) + (V ) (3.15)
where
(IV ) =
2 πiτ 22 Λ^22
dwdz L(0, 0) (3.16)
|(V )| ≤
C 3
Λ^32
‖d^3 wL(·,0)‖L^1 +‖d^3 wdz L(·,0)‖L^1 +‖d^2 wd^2 z L(·,0)‖L^1 +
dw‖d^3 wd^2 z L(w, ·)‖L^2
and C 3 depends on t, τ 1 , τ 2. Taking into account (3.14) we also obtain
|(V )| ≤
C 3
Λ^32
‖d^3 wG(·,·,·,0)‖L 1 +‖d^3 wdz G(·,·,·,0)‖L 1 +‖d^2 wd^2 z G(·,·,·,0)‖L 1
dw
dx
∫ (^) t−τ 2
−τ 2
dy ‖d^3 wd^3 z G(x,y,w,·)‖L 2
C 3
Λ^32
[ ∫ (^) τ 2
0
ds′
dξ
dη |d^3 ηG+ 12 (R, ξ, s′, η, τ 2 )| +
∫ (^) τ 2
0
ds′
dξ
dη |d^3 ηdsG+ 12 (R, ξ, s′, η, τ 2 )|
∫ (^) τ 2
0
ds′
dξ
dη |d^2 ηd^2 s G+ 12 (R, ξ, s′, η, τ 2 )| +
∫ (^) t
0
ds′
dξ
dη
(∫ (^) t
0
ds |d^3 ηd^3 s G+ 12 (R, ξ, s′, η, s)|^2
) 1 / 2 ]
17
Concerning dwdz L(0,0), a straightforward computation gives
dwdz L(0, 0) =
dx dwG(x,τ 2 − τ 1 , 0 ,0) ei^
Λ 1 τ 1 (τ^2 −τ^1 )x^ +
dx
∫ (^) τ 2 −τ 1
−τ 1
dy dwdz G(x,y, 0 ,0) ei^
Λ 1 τ 1 xy
≡ (IV 1 ) + (IV 2 ) (3.19)
In (3.19) we integrate by parts in the first integral and use (2.17), (2.20) in the second integral. Then
|(IV 1 )| ≤
τ 1 τ 2 − τ 1
‖dxdwG(·,τ 2 − τ 1 , 0 ,0)‖L 1
τ 1 τ 2 − τ 1
dξ |dξdηG+ 12 (ξ, τ 2 , q 2 , τ 2 )| (3.20)
|(IV 2 )| ≤
C 1
‖dwdz G(·, 0 , 0 ,0)‖L 1 +
dx‖dxdydwdz G(x,·, 0 ,0)‖L 2
C 1
[∫
dξ |dηdsG+ 12 (ξ, τ 1 , q 2 , τ 2 )| +
dξ
(∫ (^) τ 2
0
ds′^ |dξds′ dηdsG+ 12 (ξ, s′, q 2 , τ 2 )|^2
) 1 / 2 ]
and C 1 depends on τ 1 , τ 2. Taking into account (3.12), (3.18), (3.20), (3.21) we get (3.7), with an explicit estimate of R+ 12 (R, t, Λ 1 , Λ 2 ), and this concludes the proof.
Let us consider the cases i),ii),iii) where the stationary point of the phase lies out of the integration region. In such cases, exploiting repeated integration by parts, one can show that (3.3) is O(Λk 1 ), for any integer k, for Λ 1 → ∞. Since the error term in (3.7) is O(Λ− 1 3 ), in the next proposition we shall limit to k = 3.
Proposition 3.2. For a 2 > 0 , t > τ 2 we have
hajl(t) =
Λ^31
Rajl(·, t, Λ 1 , Λ 2 ) (3.22)
for a = ±, j = 2, l = 1 and a = −, j = 1, l = 2, where
|Rajl(R,t,Λ 1 ,Λ 2 )| ≤ C
∫ (^) t
0
ds
∫ (^) s
0
ds′
dξ
dη
∣d^3 ηGajl(R, ξ, s′,η,s)+d^3 ξ Gajl(R, ξ, s′,η,s)
and C depends on τ 1 , τ 2.
Proof. Let us define τ 0 = τ^1 + 2 τ^2 and write
19
f (^) n(2) 1 n 2 (t) =
A(2)
e−^ ℏi tK^0 ψ+ 12 +
Λ^31
R(2)(·, t, Λ 1 , Λ 2 ) (3.28)
A(2)^ = − 4 π^2
λ^2 τ 1 τ 2 ℏ^2
e−i(n^1 +n^2 +1)ωt−iΛ^1 q^1 −iΛ^2 q^2 ei^
ℏ 2 M γ^2 (τ^1 q (^21) +τ 2 q 22 +2τ 2 q 1 q 2 ) g 1 (q 1 )g 2 (q 2 ) (3.29)
ψ 12 + = ψˆ+ 2 (·, q 1 , τ 1 , q 2 , τ 2 ) (3.30)
R(2)(·, t, Λ 1 , Λ 2 ) = i
λ ℏ
Γn 1 n 2 (t)
a=± j,l=1, 2 ,j 6 =l
Rajl(·, t, Λ 1 , Λ 2 ) (3.31)
Notice that the leading term in (3.28) can also be written as
A(2)
e−^ ℏi tK^0 ψ 12 + = − 4 π^2
λ^20 δm δE
eiη^12 (t)^
j=1, 2
V^ ˜ (qj ) (˜φnj φ 0 )(qj ) e−^ ℏi^ tK^0 ψ+ 12 (3.32)
η 12 (t) =
n^21 2
δE δτ 1
n^22 2
δE δτ 2
δE δτ 2
− (n 1 + n 2 + 1)ωt +
n 1 δτ 1
n 2 δτ 2
ψ+ 12 (R) =
N
σ
e−^
(R−R 12 )^2 2 σ^2 +i^
P 12 ℏ R, R 12 = (n 1 a 1 +n 2 a 2 ) δE, P 12 = P 0 [1 − (n 1 + n 2 ) δE]
(3.34)
3.2. The case a 2 < 0.
Here the two oscillators are on the opposite sides with respect to the origin and one can easily check that the point (ξ 0 , s′ 0 , η 0 , s 0 ) where the phase in (3.3) is stationary is: (q 1 , τ 1 , q 2 , −τ 2 ) for h+ 12 (t), (−q 1 , −τ 1 , −q 2 , τ 2 ) for h− 12 (t), (q 2 , −τ 2 , q 1 , τ 1 ) for h+ 21 (t), (−q 2 , τ 2 , −q 1 , −τ 1 ) for h− 21 (t). Since none of these points belongs to the domain of integration we can show that h± jl(t) is always rapidly decreasing to zero for Λ 1 , Λ 2 → ∞.
Proposition 3.3. For a 2 < 0 , t > τ 2 and any integer k > 2 we have
h± jl(t) =
Λk 1
Q± jl(·, t, Λ 1 , Λ 2 ), j, l = 1, 2 (3.35)
where
|Q± jl(R, t, Λ 1 , Λ 2 )| ≤
∫ (^) t
0
ds
∫ (^) s
0
ds′
dξ
dη
|dkη G± jl(R, ξ, s′, η, s)| + |dkξ G± jl(R, ξ, s′, η, s)|
20 GIANFAUSTO DELL’ANTONIO, RODOLFO FIGARI, AND ALESSANDRO TETA
Proof. The proof is an immediate consequence of k integration by parts and a trivial estimate.
From the above proposition we conclude that for n 1 , n 2 6 = 0, a 2 < 0, t > τ 2 and any integer k > 2 we have
f (^) n(2) 1 n 2 (t) =
Λk 1
Q(2)(·, t, Λ 1 , Λ 2 ) (3.37)
Q(2)(·, t, Λ 1 , Λ 2 ) = i
λ ℏ
Γn 1 n 2 (t)
a=± j,l=1, 2 ,j 6 =l
Qajl(·, t, Λ 1 , Λ 2 ) (3.38)
- Joint excitation probability
We are now in position to compute the joint excitation probability of the two oscillators in the two cases a 2 < 0 and a 2 > 0. As a preliminary step, we need a pointwise estimate of the derivatives of G± jl with respect to the variables ξ, η. It is convenient to introduce the following notation
a =
ℏt M γ^2
, b =
ℏt M γσ
, c =
σ γ
s = tα, s′^ = tβ, (4.2) x = σ−^1 R, z = x + b(βξ + αη) (4.3)
We notice that, for t of the same order of magnitude of τ 2 , the constants in (4.1) are of order one; moreover the rescaled variables α, β satisfy 0 ≤ α, β ≤ 1.
Lemma 4.1. We have
|dkη G± jl(R, ξ, tβ, η, tα)| + |dkξ G± jl(R, ξ, tβ, η, tα)|
≤ c Ak(t)
N
σ
〈z〉k^ e−^
z 22 〈ξ〉k
∑k
m=
|dmξ gj (ξ)| 〈η〉k
∑k
m=
|dmη gl(η)| (4.4)
where
Ak(t) =
1 + a^2 + b^4
)k/ 2 ( 1 + b^2 + c^2
)k/ 2 (1 + a)k^ (4.5)
