Spohn H. Dynamics of Charged Particles and their Radiation Field

Подождите немного. Документ загружается.

17.5 Scattering theory 273

In particular, H has the absolutely continuous spectrum [E

, ∞) of inﬁnite multi-

plicity.

Under asymptotic completeness the long-time dynamics is fully characterized

through

Proposition 17.5 (Relaxation to the ground state). Let A be local in the sense

that A ∈ B(

) ⊗ W. Then for every ψ ∈ Ran 

−

with ψ=1 we have

lim

t→∞

e

−iHt

ψ, Ae

−iHt

ψ=ψ

, Aψ

. (17.107)

In particular, if asymptotic completeness holds, then the limit (17.107) is valid for

all ψ ∈

⊗ F.

Proof : Let ψ = 

−

φ with φ ∈ D

.By(17.101) one has

lim

t→∞

e

−iHt

ψ, Ae

−iHt

ψ= lim

t→∞

J e

−iH

φ, AJe

−iH

φ, (17.108)

which converges to the limit (17.107) as is seen by the argument explained in

(17.98). Any ψ ∈ Ran 

−

can be approximated through states of the form 

−

with φ ∈ D

. 2

Proposition 17.6 (Cook estimate). Let the integrability condition (17.27) be sat-

isﬁed. Then for all φ ∈ D

the strong limit

lim

t→∞

i(H −E

J e

−iH

φ = 

−

φ (17.109)

exists.

Proof :Ifφ = , the limit exists and is ψ

. Let then ,φ=0 and φ ∈ D

∩

D(H

). Then Je

−iH

φ ∈ D(H ) and we have

i(H −E

J e

−iH

φ = ie

i(H −E

(HJ − E

J − JH

−iH

=−e

i(H −E

eQψ

⊗

·E

−

−iH

φ. (17.110)

Here

−

= i



λ=1,2



kϕ(k)



ω(k)/2e

(k)a(k,λ) (17.111)

and we used

a( f )(ψ

⊗

φ) = a( f )ψ

⊗

φ + ψ

⊗

a( f )φ, (17.112)

∗

( f )(ψ

⊗

φ) = a

∗

( f )ψ

⊗

φ, (17.113)

274 Radiation

which follow from the deﬁnition (17.92). Thus

i(H −E

J e

−iH

φ = J φ − e



ds e

i(H −E

Qψ

⊗

·E

−

−iH

φ (17.114)

and it is to be shown that t →Qψ

⊗

·E

−

−iH

φ is integrable for a dense

set of φ’s.For this purpose we deﬁne L

⊂ L

⊥

, R

) to be the linear subspace

spanned by the set {e

−iωt

ϕ

√

ω/2e

|t ∈ R}.Wechoose an n-photon vector in prod-

uct form, φ = (0,...,φ

, 0,...),φ

= S



j=1



with each factor being a sum



(k,λ) =





=1

j

−iω(k)t

j

ϕ(k)



ω(k)/2e

(k) +



⊥

(k,λ) (17.115)

with



⊥

orthogonal to L

. Then

Qψ

⊗

·E

−

−iH

φ≤



j=1





λ=1,2



kϕ(k)



ω(k)/2e

(k) ·



(k,λ)e

−iω(k)t



(17.116)

Inserting from (17.115) yields a ﬁnite sum of terms of the form



k|ϕ(k)|

ω(k)Q

⊥

(k)e

−iω(k)(t+s)

(17.117)

which are integrable, either by assumption or as a matter of fact for the Pauli–Fierz

model, cf. the remark below Theorem 17.1.

Our argument establishes the limit (17.109) for a dense set of vectors in the

n-photon subspace. By linearity and by taking uniform limits, this extends to all of

. 2

For ψ ∈ Ran 

one has all the desired properties, relaxation to the ground

state as in Proposition 17.5, long-time asymptotics as in (17.90), and spectral mea-

sures which are absolutely continuous except for a possible mass at E

with weight

ψ, ψ

. Asymptotic completeness, i.e. the property Ran 

= C

⊗ F, ensures

that there are no states with unphysical dynamics.

Notes and references

Section 17.1

The dipole approximation in conjunction with the N -level approximation is com-

mon practice in atomic physics, for example Agarwal (1974), Cohen-Tannoudji

Notes and references 275

et al. (1992). The unitary transformation (17.7) is linked with Power and Zienau

(1959). It is also used by Bloch and Nordsieck (1937). For the special case of

N = 2 the transition from Fock to non-Fock ground state is studied in consid-

erable detail by Leggett et al. (1987), Spohn (1989), Amann (1991), and Weiss

(1999). The corresponding Hamiltonian, H

,isknown as the spin-boson model.

The vector character of the Bose ﬁeld is ignored and one sets H

= εσ

, Q · A





g(k)a

∗

(k) +



g(k)

∗

a(k)). The t

−2

-decay of (17.6) is the so-called Ohmic

case, which is marginal for the transition to non-Fock. For small coupling H

has

a unique ground state in

⊗ F, whereas for large coupling H

acquires an in-

ﬁnite number of bosons, which leads to a two-fold degenerate ground state, both

lying outside Fock space. Form factors with a decay different from t

−2

have been

also investigated.

Section 17.2

Landau (1927) uses density matrices in the description of the reduced state of

the atom. He arrives at a variant of the master equation (17.32). Its diagonal part

is often referred to as the Pauli master equation (Pauli 1928). A further inﬂuen-

tial work is Bloch (1928). The systematic weak coupling theory goes back to van

Hove (1955, 1957) and has been further developed in response to the theoreti-

cal challenges in quantum optics. Just to remind the reader: In theoretical models

of the laser one has to include dissipation for the ﬁeld modes of the cavity to

account for lossy reﬂection at the walls. For photon counting statistics one has

to devise a simple model of a detector. An interesting exchange is Srinivas and

Davies (1981) and Mandel (1981). For laser cooling and trapping the spontaneous

emission and its associated recoil must be described in a concise way (Metcalf and

van der Straten 1999). Thus the general problem of how to model open quantum

systems necessarily comes into focus. On the classical level the addition of friction

forces and possibly of noise serves well. But quantum mechanics poses constraints

which are still of current research interest. As a short sample out of a large body

of literature we refer to Lax (1968), Glauber (1969), Kossakowski (1972), Haake

(1973), Spohn (1980), Carmichael (1999), Weiss (1999), and Breuer and Petruc-

cione (2002). Our presentation here is based on Davies (1974, 1975, 1976a). He

emphasizes time-averaging which has been overlooked mostly, but is done cor-

rectly in Cohen-Tannoudji et al. (1992) and Breuer and Petruccione (2002). The

various generators of the dissipative evolution in the weak coupling limit are com-

pared in D

umcke and Spohn (1979). In the text we discussed only single-time

statistics. Stationary two-time statistics appear frequently in applications. Multi-

time statistics are studied by D

umcke (1983) within the presented framework.

276 Radiation

Although even the most simplistic theory yields a shift of the spectral line, such

predictions were not taken seriously. The rough estimate of Bethe (1947) and the

more sophisticated computation of Grotch (1981) resulting in a cutoff-independent

shift could have been done as early as 1930. It is only through the war-related re-

search on radar that experimental techniques became available to measure such

ﬁne effects. The theory followed soon; see Schweber (1994) for an excellent ac-

count.

The weak coupling theory is also a useful tool in studying decoherence. In

essence one starts the dynamics with a coherent superposition of two spatially

well-separated wave packets. According to the appropriate quantum master equa-

tion such a coherence is destroyed on a time scale which is much, much shorter

than the friction time scale. Properly speaking the master equation should not be

used on such short time scales. When decoherence is due to the coupling to the

quantized rediation ﬁeld, D

urr and Spohn (2002) provide an analysis based on

the dipole approximation. A complete discussion, avoiding the dipole approxima-

tion, is given by Breuer and Petruccione (2001), who also list references to earlier

work.

The weak coupling theory had a mathematical spin-off, going way beyond the

speciﬁc application at hand. The basic observation is that the dissipative semigroup

is the classical analog of the transition probability of a classical Markov process,

the Markov character being embodied in the semigroup property T

= T

t+s

t, s ≥ 0. T

is positivity and normalization preserving, in the sense that if ρ is a

density matrix so is T

ρ.Asrecognized by Lindblad (1976) the stronger notion of

complete positivity is very natural. It means that if

H is extended to H ⊗ C

and

in the trivial way to T

⊗ 1, then T

⊗ 1ispositivity preserving for every n.In

this framework the possible types of generators are classiﬁed by Lindblad (1976).

He also characterizes dissipation through the decrease of relative entropy (Lind-

blad 1975). Mixing and the long-time limit t →∞are studied by Spohn (1976),

Frigerio (1978), and Frigerio and Verri (1982). The generalization of the notion

of detailed balance to the quantum context is discussed by Gorini et al. (1984).

Most recommended introductions are Davies (1976b) and Alicki and Lendi (1987).

Clearly the next level is to inquire about multitime statistics and their build-up

from the semigroup T

. This is a fairly straightforward step for classical Markov

processes through the concept of conditional independence of past and future. No

such thing seems to exist on the quantum level and the theory of quantum stochas-

tic processes tries to provide a consistent framework, possibly guided by speciﬁc

model systems, that can be analyzed in detail. We refer to Accardi, Frigerio and

Lewis (1982), Lindblad (1983), Hudson and Parthasarathy (1984), Accardi et al.

(1991), and Parthasarathy (1992), and the recent monographs by Alicki and Fannes

(2001) and by Accardi et al. (2001).

Notes and references 277

Section 17.3

Already in his ﬁrst work on radiation theory Dirac (1927) simpliﬁes the problem

to a single level coupled to a continuum of modes. A two-level atom coupled to

the radiation ﬁeld in the rotating wave approximation also reduces to a Friedrich–

Lee-type Hamiltonian.

Complex dilations were investigated in connection with the study of Regge

poles, cf. Reed and Simon (1978) for references. The mathematical framework

is developed by Aguilar and Combes (1971) and Balslev and Combes (1971). A

beautiful survey is Simon (1978). For an introduction we refer to Cycon et al.

(1987). Hunziker (1990) focuses on the question of how to translate the results on

the resolvent back to the real time-domain. Okamoto and Yajima (1985) observe

that the dilation of the massive photon ﬁeld can be used to unfold resonances. Res-

onances of the Pauli–Fierz model are studied in Bach, Fr

ohlich and Sigal (1995,

1998a, 1998b, 1999). They develop a renormalization-type iterative procedure to

pin down the domain of analyticity of the complex dilated resolvent. This method

is reﬁned by Bach et al. (2002). An inﬁnitesimal version based on Mourre-type

estimates and the Feshbach method is Derezi

nski and Jak

c (2001).

Section 17.4

Our discussion is based on Davies (1976a). L

is the Davies generator in the weak

coupling theory. Line shapes are discussed by Weisskopf and Wigner (1930). Our

examples for the spectral characteristics of the emitted light are taken from Cohen-

Tannoudji et al. (1992), Chapter IIIC and Exercise 15.

Section 17.5

Potential scattering is discussed in Reed and Simon (1979) and N -body scatter-

ing in Cycon et al. (1987). We follow the presentation in H

ubner and Spohn

(1995a). The Cook argument is based on Høegh-Krohn (1970) who also studies

the asymptotic electromagnetic ﬁelds; for a complete discussion see (Fr

ohlich,

Griesemer and Schlein 2001). In the meantime the mathematical investigation of

scattering of photons from an atom has ﬂourished. For simplicity often the scalar

ﬁeld model of section 19.2 is studied. An important step is Derezi

nski and G

erard

(1999) who establish asymptotic completeness in the case of massive photons,

ω(k) = (k

+ m

)

1/2

, m

> 0, and a strictly conﬁning potential. Earlier work

restricted to an N -level atom is G

erard (1996) and Skibsted (1998). An extension

to massless photons under the condition ϕ(0) = 0isG

erard (2002). For m

> 0

ohlich, Griesemer and Schlein (2001, 2002) allow for potentials which are not

278 Radiation

strictly conﬁning, like the Coulomb potential. Thereby the channel for a freely

propagating electron is opened up as it occurs in the description of the photoelec-

tric effect (Bach, Klopp and Zenk 2002). Ammari (2000) establishes asymptotic

completeness for the Nelson model of section 19.2 with ultraviolet cutoff removed.

In these works asymptotic completeness is deﬁned in such a way that H could

have other eigenvalues besides its ground state. To exclude them one has to resort

to Bach, Fr

ohlich and Sigal (1998a) and Fr

ohlich, Griesemer and Schlein (2002).

Adifferent approach is to take the dipole approximation with harmonic con-

ﬁning potential. Since the Hamiltonian is quadratic, the scattering theory can be

reduced to one-particle scattering with a ﬁnite rank perturbation. Maassen (1984)

notices that for a weakly anharmonic conﬁning potential the time-dependent

perturbation series can be controlled uniformly in time. His estimates are improved

and optimized in Maassen, Gut

a and Botvich (1999) and Fidaleo and Liverani

(1999). With this input one can prove asymptotic completeness in the strong sense

of Deﬁnition 17.4. The perturbing potential must be bounded and so small that the

conﬁning potential remains convex (Spohn 1997). The harmonic case is investi-

gated by Arai (1983b).

Relaxation at ﬁnite temperatures

The weak coupling theory of chapter 17 is the workhorse of quantum optics and

serves very well in practice, also at nonzero temperatures. From the viewpoint of

the theory one might wonder about the structure at ﬁxed small, but nonzero, cou-

pling strength, which needs to go beyond the analysis of the weak coupling theory.

Much effort has been invested to achieve this goal, basically by trying to iden-

tify corrections within time-dependent perturbation theory. Unfortunately, since

the long-time behavior must be extracted, the details quickly become unwieldy

and one has to rely on ad hoc approximations.

Over recent years a novel approach has been pursued which investigates the

pole structure of the analytic continuation of the resolvent of H

across the real

axis through complex dilations; compare with section 17.3. The techniques are

demanding but simplify substantially at ﬁnite temperatures when the Hamiltonian

is replaced by the Liouvillean and, since its spectrum is the full real line, complex

dilations are replaced by complex translations which can be handled more easily.

From the pole structure a fairly complete picture of the long-time dynamics can be

extracted with the potential of computing systematically higher corrections to the

weak coupling theory.

The ﬁnite temperature relaxation is a digression into the realm of time-

dependent statistical mechanics with small deviations from thermal equilibrium.

While this is of independent interest and has important applications in quantum

optics and condensed matter, our goal is merely to illustrate the power of complex

translations and make the connection to the weak coupling theory.

For completeness we recall once again the set-up. In the dipole approximation

and N -level approximation the Hamiltonian is

= H

+ H

+ λQ · E

= H

+ λH

int

, (18.1)

see (17.10). H

acts on C

. Diverging from our previous convention, the energies

are labeled as ε

≤ ε

... ≤ ε

allowing for possible degeneracies. Q is the

279

280 Relaxation at ﬁnite temperatures

dipole operator as an N × N matrix. The ﬁeld energy H

, and the electric ﬁeld

= E

(0) are operators on Fock space F.For ﬁnite temperatures we need some

extra structure, which will be explained below.

The photon ﬁeld is at temperature T > 0. It will be more convenient to work

with the inverse temperature and set β = 1/k

T .Inthe initial state the atom and

its nearby photons are out of equilibrium and the goal is to understand how the

coupled system relaxes back to global equilibrium. In the weak coupling theory

one disentangles the effective dynamics of the atom and regards the ﬁeld as driven

by the atomic source. At ﬁxed λ such a distinction becomes hazy and a more global

view is adopted with the separation deduced in the small-λ limit.

The analysis of thermal relaxation relies on the following strategy. One intro-

duces coordinates which encode the ﬁnite energy excitations away from equi-

librium. Doing this properly relies on tools from the representation theory of

∗

-algebras which for our context was mostly developed in the 1960–70s. For

noninteracting photons the representation of Araki and Woods (1963) is of a suf-

ﬁciently concrete form and also allows the incorporation of the coupling to the

atom. Note that at zero coupling the spectrum of ﬁnite energy excitation covers

the full real axis

R, since ω(k) =|k| and energy can be either below or above its

equilibrium value. The energy differences of the atom are embedded in this spec-

trum as discrete eigenvalues. As the coupling is turned on, they become resonances

which are uncovered by a complex downward translation of the photon excitation

spectrum. The location of the resonance poles and the corresponding eigenspaces

can be handled through standard analytic perturbation theory.

To convince the reader that the Araki–Woods Liouvillean correctly describes the

ﬁnite energy excitation, we need some background material on quantum systems

at ﬁnite temperature. We conform with established notation which to some extent

deviates from our previous conventions.

18.1 Bounded quantum systems, Liouvillean

We start with an abstract quantum system on a separable Hilbert space

H equipped

with the scalar product ·, ·.Weassume that the Hamiltonian H is bounded from

below and has a purely discrete spectrum such that

tr[e

−β H

] < ∞ (18.2)

for arbitrary β>0. The algebra of observables,

A,isthe set of all bounded op-

erators on

H, denoted by B(H).Ageneral quantum state is given through the

density matrix ρ, satisfying ρ ≥ 0, trρ = 1. In particular ρ ∈

(H), denoting the

two-sided ideal of trace class operators on

H.Inthe Heisenberg picture the time

18.1 Bounded quantum systems, Liouvillean 281

evolution is given through

(a) = e

itH

−itH

(18.3)

as acting on a ∈ B(

H). The dual Schr

odinger picture provides the time evolution

of states as

ρ → ρ

= α

−t

(ρ) = e

−itH

ρe

itH

. (18.4)

We want the evolution of density matrices to look like the evolution of vectors

on a Hilbert space and, for this purpose, introduce the two-sided ideal of Hilbert–

Schmidt operators

(H).Abounded operator a belongs to T

(H) if and only if

tr[a

∗

a] < ∞. T

(H) becomes a Hilbert space under the scalar product

a|b=tr[a

∗

b], a, b ∈ T

(H). (18.5)

It will be useful to represent

A = B(H) as an algebra of operators on T

(H).

(H) carries a left representation through

(a)κ = aκ ∈

(H). (18.6)

Later on we will need also the right antirepresentation deﬁned through

r(a)κ = κa

∗

∈ T

(H). (18.7)

This representation is antilinear since r(za)κ = z

∗

r(a)κ for z ∈ C.

We transcribe states and dynamics to

(H).Toevery element κ ∈ T

(H) a

state ρ is associated through

ρ =κ|κ

−1

κκ

∗

. (18.8)

The expectation of a ∈ B(

H) is given by

a

= tr[ρa] =κ|κ

−1

κ|(a)κ. (18.9)

The time evolution becomes

α

(a)

=κ|κ

−1

tr[κ

∗

(a)κ] =κ|κ

−1

κ|(α

(a))κ (18.10)

and for κ, σ ∈

(H)

κ|(α

(a))σ =tr[κ

∗

(a)σ ] = tr[(e

−itH

κe

itH

)

∗

a(e

−itH

σ e

itH

)]

=α

−t

(κ)|(a)α

−t

(σ )=e

−itL

κ|(a)e

−itL

κ. (18.11)

The last identity deﬁnes the Liouvillean

L. Clearly

Lκ = [H,κ] (18.12)

282 Relaxation at ﬁnite temperatures

and κ

= α

−t

(κ) is governed by the Schr

odinger-like equation

= Lκ

. (18.13)

The Liouvillean is a symmetric operator as can be seen from

κ|

Lσ =tr[κ

∗

[H,σ]] = tr[



[H,κ]



∗

σ ] =Lκ|σ . (18.14)

To work concretely with the left, respectively right, representation of

A and the

Liouvillean

L it is convenient to identify T

(H) with H ⊗ H through the isomor-

phism

: T

(H) → H ⊗ H. (18.15)

In a suitable basis C is simply complex conjugation. More abstractly C is an anti-

unitary involution on

H, i.e.

= 1 and Cψ, Cϕ=ϕ, ψ. (18.16)

Then with κ deﬁned through κψ = ψ

ψ

,ψ one sets

κ = ψ

⊗ Cψ

∈ H ⊗ H (18.17)

which extends by linearity. Note that

(a)κ = I

aκ = aψ

⊗ Cψ

= (a ⊗ 1)I

κ. (18.18)

Thus I

intertwines with the left representation  of A on H ⊗ H given by

(a) = a ⊗ 1. (18.19)

Similarly

r(a)κ = I

(κa

∗

) = ψ

⊗ Caψ

= 1 ⊗ CaCI

κ (18.20)

and

r(a) = 1 ⊗CaC on

H ⊗ H. (18.21)

In particular for the Liouvillean

Lκ = H ψ

⊗ Cψ

− ψ

⊗ CHψ

= H ψ

⊗ Cψ

− ψ

⊗ (CHC)Cψ

= (H ⊗ 1 − 1 ⊗ CHC)(ψ

⊗ Cψ

) = (H ⊗ 1 − 1 ⊗ CHC)I

κ (18.22)

and

L = I

−1

= H ⊗ 1 − 1 ⊗ CHC on H ⊗ H. (18.23)