Reduced and Extended Weak Coupling Limit
****************************************
BANACH CENTER PUBLICATIONS, VOLUME **
INSTITUTE OF MATHEMATICS
POLISH ACADEMY OF SCIENCES
WARSZAWA 200*
REDUCED AND EXTENDED
WEAK COUPLING LIMIT
JAN DEREZIŃSKI
Department of Mathematical Methods in Physics
Warsaw University
Hoża 74, 00-682, Warszawa, Poland
E-mail: jan.derezinski@fuw.edu.pl
WOJCIECH DE ROECK
Instituut voor Theoretische Fysica, K.U.Leuven
Belgium
E-mail: wojciech.deroeck@fys.kuleuven.be
Contents 1
1 Introduction. 2
2 Toy model of the weak coupling limit. 4
2.1 Dilations of contractive semigroups. . . . . . . . . . . . . . . . . . . . . . 4
2.2 “Toy quadratic noises”. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Weak coupling limit for Friedrichs operators. . . . . . . . . . . . . . . . . 8
3 Completely positive maps and semigroups. 10
3.1 Completely positive maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Completely positive semigroups. . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Classical Markov semigroups. . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4 Invariant c.p semigroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.5 Detailed Balance Condition. . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Bosonic reservoirs. 18
4.1 Second quantization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Coupling to a bosonic reservoir. . . . . . . . . . . . . . . . . . . . . . . . . 19
The paper is in final form and no version of it will be published elsewhere.
http://arxiv.org/abs/0704.0669v2
2 J. DEREZIŃSKI AND W. DE ROECK
4.3 Thermal reservoirs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5 Quantum Langevin dynamics. 21
5.1 Linear noises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.2 Quadratic noises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.3 Total energy operator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6 Weak coupling limit for Pauli-Fierz operators. 25
6.1 Reduced weak coupling limit. . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.2 Energy of the reservoir in the weak coupling limit. . . . . . . . . . . . . . 27
6.3 Extended weak coupling limit. . . . . . . . . . . . . . . . . . . . . . . . . 27
1. Introduction. Physicists often describe quantum systems by completely positive (c.p.)
semigroups [Haa, AL, Al2]. It is generally believed that this approach is only a phe-
nomenological approximation to a more fundamental description. One usually assumes
that on the fundamental level the dynamics of quantum systems is unitary, more precisely,
is of the form t 7→ eitH · e−itH for some self-adjoint H .
One of justifications for the use of c.p. semigroups in quantum physics is based on
the so-called weak coupling limit for the reduced dynamics [VH, Da1], which we will call
the reduced weak coupling limit. One assumes that a small system is coupled to a large
reservoir and the dynamics of the full system is unitary. The interaction between the
small system and the reservoir is multiplied by a small coupling constant λ. Often the
reservoir is described by a free Bose gas.
The basic steps of the reduced weak coupling limit are:
• Reduce the dynamics to the small system.
• Rescale time as t
• Subtract the dynamics of the small system.
• Consider the weak coupling λ→ 0.
In the limit one obtains a dynamics given by a c.p. Markov semigroup.
Another possible justification of c.p. dynamics goes as follows. One considers the
tensor product of the small system and an appropriate bosonic reservoir. On this enlarged
space one constructs a certain unitary dynamics whose reduction to the small system is
a c.p. semigroup. We will call it a quantum Langevin dynamics. Another name used in
this context in the literature is a quantum stochastic dynamics. Its construction has a
long history, let us mention [AFLe, HP, Fr, Maa].
Thus one can obtain a Markov c.p. semigroup by reducing a single unitary dynamics,
without invoking a family of dynamics and taking its limit. However, the generator of a
quantum Langevin dynamics equals i[Z, ·] where Z is a self-adjoint operator that does
not look like a physically realistic Hamiltonian. In particular, it is unbounded from both
below and above. Thus one can question the physical relevance of this construction.
REDUCED AND EXTENDED WEAK COUPLING LIMIT 3
It turns out, however, that one can extend the weak coupling limit in such a way, that
it involves not only the small system, but also the reservoir. As a result of this approach
one can obtain a quantum Langevin dynamics. One can argue that this approach gives
a physical justification of quantum Langevin dynamics.
The above idea was first implemented in [AFL] by Accardi, Frigerio and Lu under
the name of the stochastic limit (see also [ALV]). Recently we presented our version of
this approach, under the name of the extended weak coupling limit [DD1, DD2], which
we believe is simpler and more natural that of [AFL]. The basic steps of the extended
weak coupling limit are:
• Introduce the so-called asymptotic space — the tensor product of the space of the
small system and of the asymptotic reservoir.
• Introduce an identification operator that maps the asymptotic reservoir into the
physical reservoir and rescales its energy by λ2 around the Bohr frequencies.
• Rescale time as t
• Subtract the “fast degrees of freedom”.
• Consider the weak coupling λ→ 0.
In the limit one obtains a quantum Langevin dynamics on the asymptotic space. Note
that the asymptotic reservoir is given by a bosonic Fock space (just as the physical
reservoir). Its states are however different – correspond only to those physical bosons
whose energies differ from the Bohr frequencies by at most O(λ2). Only such bosons
survive the weak coupling limit.
Let us mention yet another scheme of deriving quantum Langevin equations that has
received attention in the literature, namely the ‘repeated interaction models’ where the
reservoir is continuously refreshed, see [AtP, AtJ].
In this article we review various aspects of the weak coupling limit, reduced and, es-
pecially, extended, mostly following our papers [DD1, DD2]. We also describe some back-
ground material, especially related to completely positive semigroups, quantum Langevin
dynamics and the Detailed Balance Condition.
The plan of our article is as follows. In Section 2 we describe both kinds of the weak
coupling limit on a class of toy-examples – the so-called Friedrichs Hamiltonians and
their dilations. They are less relevant physically than the main model treated in our
article – the one based on Pauli-Fierz operators. Nevertheless, they illustrate some of the
main ideas of this limit in a simple and mathematically instructive context. This section
is based on [DD1].
In Sections 3 we recall some facts about completely positive maps and semigroups,
sketching proofs of the Stinespring dilation theorem [St] and of the so-called Lindblad
form of the generator of a c.p. semigroup [Li, GKS]. In particular, we discuss the freedom
of choosing various terms in the Lindblad form. This question, which we have not seen
discussed in the literature, is relevant for the construction of quantum Langevin dynamics
and the weak coupling limit.
4 J. DEREZIŃSKI AND W. DE ROECK
C.p. semigroups that arise in the weak coupling limit have an additional property –
they commute with the unitary dynamics generated by the Hamiltonian K of the small
system – for shortness we say that they are K-invariant. If in addition the reservoir is
thermal, they satisfy another special property – the so-called Detailed Balance Condition
(DBC) [DF1, Ag, FKGV, Al1]. We devote a large part of Sect. 3 to an analysis of the
K-invariance and the DBC. We show that the generator of a c.p. semigroup with these
properties has some features that curiously resemble the Tomita-Takesaki theory and the
KMS condition. Let us note that in our article the DBC is considered jointly with the
K-invariance, because c.p. semigroups obtained in the weak coupling limit always have
the latter property.
In Section 4 we describe the terminology and notation that we use to describe second-
quantized bosonic reservoirs interacting with a small system. In particular, we introduce
Pauli-Fierz operators [DJ1] – used often (also under other names) in the physics literature
to describe physically realistic systems. In Subsect. 4.3 we discuss thermal reservoirs. In
our definition of a thermal reservoir at inverse temperature β one needs to check a simple
condition for the interaction without explicitly invoking the concept of a KMS state on
an operator algebra, or of a thermal Araki-Woods representation of the CCR [DF1, DJ1].
Of course, this condition is closely related to the KMS property.
In Subsection 5 we describe a construction of quantum Langevin dynamics. We include
a discussion of the so-called quadratic noises, even though they are still not used in our
version of the extended weak coupling limit. (See however [Go] for some partial results
in the context of the formalism of [AFL]).
In Section 6 we describe the two kinds of the weak coupling limit for Pauli-Fierz
operators: reduced and extended. Most of this section is based on [DD2].
2. Toy model of the weak coupling limit. This section is somewhat independent of
the remaining part of the article. It explains the (reduced and extended) weak coupling
limit in the setting of contractive semigroups on a Hilbert space and their unitary dila-
tions. It gives us an opportunity to explain some of the main ideas of the weak coupling
limit in a relatively simple setting. It is based on [DD1].
It is possible to construct physically interesting models based on the material of this
section (e.g. by considering quadratic Hamiltonians obtained by second quantization).
We will not discuss this possibility further, since in the remaining part of the article we
will analyze more interesting and more realistic models.
2.1. Dilations of contractive semigroups. First let us recall the well known concept of
a unitary dilation of a contractive semigroup. Let K be a Hilbert space and e−itΥ a
strongly continuous contractive semigroup on K. This implies that −iΥ is dissipative:
−iΥ + iΥ∗ ≤ 0.
Let Z be a Hilbert space containing K, IK the embedding of K in Z and Ut a unitary
group on Z. We say that (Z, IK, Ut) is a dilation of e−itΥ iff
I∗KUtIK = e
−itΥ, t ≥ 0.
It is well known that every weakly continuous contractive semigroup possesses a
REDUCED AND EXTENDED WEAK COUPLING LIMIT 5
unitary dilation (which is unique up to the unitary equivalence if we additionally demand
its minimality). The original and well known construction of a unitary dilation is due
to Foias and Nagy and can be found in [NF] (see also [EL]). Below we present another
construction, which looks different from that of Foias-Nagy. Its main idea is to view the
generator of a unitary dilation as a kind of a singular Friedrichs operator. (See the next
section, where Friedrichs operators are introduced). Such a definition is well adapted
to the extended weak coupling limit. The construction that we present seems to be
less known in the mathematics literature than that of Foias-Nagy. Nevertheless, similar
constructions are scattered in the literature, especially in physics papers.
Let h be an auxiliary space and ν ∈ B(K, h) satisfy
(Υ −Υ∗) = −ν∗ν. (2.1)
Note that such h and ν always exist. One of possible choices is to take h := K and
i(Υ−Υ∗).
If φ is a vector, then |φ) will denote the operator C ∋ λ 7→ |φ)λ := λφ. Similarly, (φ|
will denote its adjoint: f 7→ (φ|f := (φ|f) ∈ C.
We will use a similar notation also for unbounded functionals. For instance, (1| will
denote the (unbounded) linear functional on L2(R) given by
(1|f =
f(x)dx (2.2)
with the domain L2(R) ∩ L1(R). |1) will denote the hermitian conjugate of (1| in the
sense of sesquilinear forms: if f ∈ L2(R) ∩ L1(R), then
(f |1) :=
f̄(x)dx.
Let ZR be the operator of multiplication on L
2(R) by the variable x.
Introduce the Hilbert spaces ZR; = h⊗L2(R) andZ := K ⊕ZR. Clearly,K is contained
in Z, so we have the obvious embedding IK : K → Z. We also have the embedding
IR : ZR → Z.
For t ≥ 0, consider the following sesquilinear form on K ⊕ (h⊗ (L2(R) ∩ L1(R))):
Ut = IRe
−itZRI∗R + IKe
−itΥI∗K (2.3)
−i(2π)− 12 IK
du e−i(t−u)Υν∗ ⊗ (1|e−iuZRI∗R
−i(2π)− 12 IR
du e−i(t−u)ZRν ⊗ |1)e−iuΥI∗K
−(2π)−1IR
0≤u1,u2, u1+u2≤t
du1du2 e
−iu2ZRν ⊗ |1)e−i(t−u2−u1)Υν∗ ⊗ (1|e−iu1ZRI∗R.
By a straightforward computation we obtain [DD1]
Theorem 2.1. The form Ut extends to a strongly continuous unitary group and
I∗KUtIK = e
−itΥ, t ≥ 0.
Thus (Z, IK, Ut) is a dilation of e−itΥ.
6 J. DEREZIŃSKI AND W. DE ROECK
Let −iZ denote the generator of Ut, so that Ut = e−itZ . Z is a self-adjoint operator
with a number of interesting properties. It is not easy to describe it with a well-defined
formula. Formally it is given by the sesquilinear form
(Υ + Υ∗) (2π)−
2 ν∗ ⊗ (1|
(2π)−
2 ν ⊗ |1) ZR
. (2.4)
Note that (2.4) looks like a special case of a Friedrichs operator (see Subsection 2.3
and [DF2]). As it stands, however, (2.4) does not define a unique self-adjoint operator.
Nevertheless, we will sometimes use the expression (2.4) when referring to Z.
Note that it is possible to give a compact formula for the resolvent of Z, (which is
another possible method of defining Z). For z ∈ C+,
(z − Z)−1 := IR(z − ZR)−1I∗R + IK(z −Υ)−1I∗K
+(2π)−
2 IK(z −Υ)−1ν∗ ⊗ (1|(z − ZR)−1I∗R
+(2π)−
2 IR(z − ZR)−1ν ⊗ |1)(z −Υ)−1I∗K
+(2π)−1IR (z − ZR)−1ν ⊗ |1) (z −Υ)−1 ν∗ ⊗ (1|(z − ZR)−1 I∗R.
Yet another approach that allows to define Z involves a “cut-off procedure”. In fact,
Z is the norm resolvent limit for r → ∞ of the following regularized operators:
Zr :=
(Υ + Υ∗) (2π)−
2 ν∗ ⊗ (1|1[−r,r](ZR)
(2π)−
2 ν ⊗ 1[−r,r](ZR)|1) 1[−r,r](ZR)ZR
Note that it is important to remove the cut-off in a symmetric way. If we replace [−r, r]
with [−r, ar] we usually obtain a different operator. The convergence of Zr to Z is the
reason why we can treat (2.4) as the formal expression for Z.
Next, let us mention a certain invariance property of Z. For λ ∈ R, introduce the
following unitary operator on Z
jλu = u, u ∈ K; jλg(y) := λ−1g(λ−2y), g ∈ ZR.
Note that
j∗λZRjλ = λ
2ZR, j
λ|1) = λ|1).
Therefore, the operator Z is invariant with respect to the following scaling:
Z = λ−2j∗λ
(Υ + Υ∗) λ(2π)−
2 ν∗ ⊗ (1|
λ(2π)−
2 ν ⊗ |1) ZR
jλ. (2.5)
(2.5) will play an important role in the extended weak coupling limit.
Note that in the weak coupling limit it is convenient to use the representation of
ZR as a multiplication operator. Another natural possibility is to represent it as the
differentiation operator. Let us describe this alternative version of the dilation.
The (unitary) Fourier transformation on h⊗L2(R) will be denoted as follows:
Ff(τ) := (2π)−1/2
f(x)e−iτxdx. (2.6)
REDUCED AND EXTENDED WEAK COUPLING LIMIT 7
We will use τ as the generic variable after the application of F . The operator Z trans-
formed by 1K ⊕F will be denoted
Ẑ := (1K ⊕ F)Z(1K ⊕F∗). (2.7)
Introduce
Dτ :=
∂τ . (2.8)
Let (δ0| have the meaning of an (unbounded) linear functional on L2(R) with the domain,
say, the first Sobolev space H1(R), such that
(δ0|f) := f(0). (2.9)
Similarly, |δ0) let be its hermitian adjoint in the sense of forms. By applying the Fourier
transform to (2.4), we can write
(Υ + Υ∗) ν∗ ⊗ (δ0|
ν ⊗ |δ0) Dτ
. (2.10)
Clearly, e−itẐ is also a dilation of e−itΥ.
The operator Ẑ (or Z) and the unitary group it generates has a number of curious
and confusing properties. Let us describe one of them. Consider the space D := K ⊕
(h⊗H1(R)). Clearly, it is a dense subspace of Z. Let us define the following quadratic
form on D:
Ẑ+ :=
Υ ν∗ ⊗ (δ0|
ν ⊗ |δ0) Dτ
. (2.11)
Then, for ψ, ψ′ ∈ D,
(ψ|(e−itẐ − 1)ψ′) = −i(ψ|Ẑ+ψ′). (2.12)
One could think that Ẑ+ = Ẑ. But Ẑ+ is in general non-self-adjoint, which is incompat-
ible with the fact that e−itẐ is a unitary group.
To explain this paradox we notice that (ψ|e−itẐψ′) is in general not differentiable at
zero: its right and left derivatives exist but are different. Hence D is not contained in the
domain of the generator of Ẑ. We will call Ẑ+ the false form of the generator of eitẐ .
In order to make an even closer contact with the usual form of the quantum Langevin
equation [HP, At, Fa, Bar, Me], define the cocycle unitary
Ŵ (t) := eitDτ e−itẐ . (2.13)
Then for t > 0, or for t = 0 and the right derivative, we have the “toy Langevin
(stochastic) equation” which holds in the sense of quadratic forms on D,
Ŵ (t) = (Υ + ν ⊗ |δt))Ŵ (t) + ν∗⊗(δt|. (2.14)
2.2. “Toy quadratic noises”. The formula for Ẑ or for Ẑ+ has one interesting feature:
it involves a perturbation that is localized just at τ = 0. One can ask whether one
can consider other dilations with more general perturbations localized at τ = 0. In this
subsection we will describe such dilations. This construction will not be needed in the
present version of the weak coupling limit. We believe it is an interesting “toy version”
8 J. DEREZIŃSKI AND W. DE ROECK
of “quadratic noises”, which we will discuss in Subsect 5.2. We also expect to extend the
results of [DD1] to “toy quadratic noises”.
Clearly, for any unitary operator U on h⊗L2(R), (1K ⊕U)eitZ(1K ⊕U∗) is a dilation
of e−itΥ. Let us choose a special U , which will lead to a perturbation localized at τ = 0.
Let S be a unitary operator on h. For ψ ∈ h⊗L2(R) ≃ L2(R, h) we set
γ(S)ψ(τ) :=
Sψ(τ), τ > 0,
ψ(τ), τ ≤ 0. (2.15)
Then γ(S) is a unitary operator on h⊗L2(R). Set
ẐS := (1K ⊕ γ(S)∗)Ẑ(1K ⊕ γ(S)).
Clearly, eitẐS is a dilation of e−itΥ. It is awkward to write down a formula for ẐS in the
matrix form, even just formally. It is more natural to write down the “false form of ZS”:
Ẑ+S := (1K ⊕ γ(S)
∗)Ẑ+(1K ⊕ γ(S))
Υ ν∗S ⊗ (δ0|
ν ⊗ |δ0) Dτ + i(1− S)⊗|δ0)(δ0|
For ψ, ψ′ ∈ D we have
(ψ|(e−itẐS − 1)ψ′) = −i(ψ|Ẑ+S ψ
′), (2.16)
Again, as in (2.14), one can extend this formula to derivatives at t > 0. Let
ŴS(t) := e
itDτ e−itẐS , (2.17)
then, in the sense of quadratic forms on D,
ŴS(t) = (Υ + ν ⊗ |δt))ŴS(t) + ν∗S⊗(δt|+ i(1 − S)⊗|δt)(δt|. (2.18)
2.3. Weak coupling limit for Friedrichs operators. Let H := K⊕HR be a Hilbert space,
where K is finite dimensional. Let IK be the embedding of K in H. Let K be a self-adjoint
operator on K and HR be a self-adjoint operator on HR. Let V be a linear operator from
K to HR. The following class of operators will be called Friedrichs operators:
Hλ :=
K λV ∗
λV HR
Assume that
‖V ∗e−itHRV ‖dt < ∞. Then we can define the following operator,
sometimes called the Level Shift Operator, since it describes the shift of eigenvalues of
Hλ in perturbation theory at the 2nd order in λ:
k∈spK
1k(K)V
∗e−it(HR−k)V 1k(K)dt, (2.19)
where 1k(K) denotes the spectral projection of K onto the eigenvalue k; spK denotes
the spectrum of K. Note that ΥK = KΥ.
The following theorem is a special case of a result of Davies [Da1, Da2, Da3], see also
[DD1]:
REDUCED AND EXTENDED WEAK COUPLING LIMIT 9
Theorem 2.2 (Reduced weak coupling limit for Friedrichs operators).
eitK/λ
−itHλ/λ
IK = e
−itΥ.
In order to study the extended weak coupling limit for Friedrichs operators we need
to make additional assumptions. They are perhaps a little complicated to state, but they
are satisfied in many concrete situations.
Assumption 2.3. We suppose that for any k ∈ spK there exists an open Ik ⊂ R and a
Hilbert space hk such that k ∈ Ik,
Ran1Ik(HR) ≃ hk ⊗ L2(Ik, dx),
1Ik(HR)HR is the multiplication operator by the variable x ∈ Ik and
1Ik(HR)V ≃
v(x)dx.
We assume that Ik are disjoint for distinct k and the measurable function Ik ∋ x 7→
v(x) ∈ B(K, hk) is continuous at k.
In other words, we assume that the reservoir Hamiltonian HR and the interaction V
are “nice” around the spectrum of K. In fact, in the extended weak coupling limit only
a vicinity of spK matters.
We set h := ⊕
hk, ZR := h⊗L2(R) and Z := K ⊕ ZR. ZR and Z are the so-called
asymptotic spaces, which are in general different from the physical spaces HR and H.
Next, let us describe the asymptotic dynamics. Let ν : K → h be defined as
ν := (2π)
v(k)1k(K).
Note that it satisfies (2.1) with Υ defined by (2.19). This follows by extending the inte-
gration in (2.19) to R and using the inverse Fourier transform. As before, we set ZR to
be the multiplication by x on L2(R) and we define e−itZ by (2.3), so that (Z, IK, e−itZ)
is a dilation of e−itΥ.
Finally, we need an identification operator that maps the asymptotic space into the
physical space. This is the least canonical part of the construction. In fact, there is some
arbitrariness in its definition for the frequencies away from spK. For λ > 0, we define
the family of partial isometries Jλ,k : L
2(R, hk) → L2(Ik, hk) ⊂ H:
(Jλ,kgk)(y) =
), if y ∈ Ik;
0, if y ∈ R\Ik.
We set Jλ : Z → H, defined for g = (gk) ∈ ZR by
Jλg :=
Jλ,kgk,
and on K equal to the identity. Note that Jλ are partial isometries and
s− lim
J∗λJλ = 1.
The following result is proven in [DD1]:
Theorem 2.4 (Extended weak coupling limit for Friedrichs operators).
s∗ − lim
iλ−2tH0eλ
−2(t−t0)Hλeiλ
−2t0H0Jλ = e
itZRe−i(t−t0)Zeit0ZR .
10 J. DEREZIŃSKI AND W. DE ROECK
Here we used the strong* limit: s∗ − limλց0 Aλ = A means that for any vector ψ we
have limλց0Aλψ = Aψ, limλց0 A
λψ = A
Note that in the extended weak coupling limit for Friedrichs operators the asymptotic
space is a direct sum of parts belonging to various eigenvalues of K that “do not talk to
one another”–have independent asymptotic dynamics.
3. Completely positive maps and semigroups. This section presents basic material
about completely positive maps and semigroups. In particular, we describe a construction
of the Stinespring dilation [St] and of the so-called Lindblad form of the generator of a c.p.
semigroup [Li, GKS]. These beautiful classic results are described in many places in the
literature. Nevertheless, some of their aspects, mostly concerning the freedom of choice
of various terms in the Lindblad form, are difficult to find in the literature. Therefore,
we describe this material at length, including sketches of proofs.
In Subsect. 3.3 we recall the usual concept of a (classical) Markov semigroups (on a
finite state space). When discussing c.p. (quantum) Markov semigroups, it is useful to
compare it to their classical analogs, which are usually much simpler.
In Subsect 3.4 we discuss c.p. semigroups invariant with respect to a certain unitary
dynamics. Such c.p. semigroups arise in the weak coupling limit – therefore, one can
argue that they are “more physical than others”.
Finally, in Subsect. 3.5 we analyze the Detailed Balance Condition, which singles out
c.p. dynamics obtained from a thermal reservoir.
3.1. Completely positive maps. Let K1,K2 be Hilbert spaces. We say that a map Ξ :
B(K1) → B(K2) is positive iff A ≥ 0 implies Ξ(A) ≥ 0. We say that Ξ is Markov iff
Ξ(1) = 1. We say that a map Ξ is n-positive iff
Ξ⊗ id : B(K1 ⊗ Cn) → B(K2 ⊗ Cn)
is positive. (id denotes the identity). We say that Ξ is completely positive, or c.p. for
short, iff it is n-positive for any n.
It is easy to see that if h be a Hilbert space and ν ∈ B(K2,K1 ⊗ h). Then
Ξ(A) := ν∗ A⊗1 ν (3.1)
is c.p. The following theorem says that the above representation of a c.p. map is universal.
2) means that this representation is unique up to a unitary isomorphism.
Theorem 3.1 (Stinespring). Assume that K1,K2 are finite dimensional.
1) If Ξ is c.p. from B(K1) to B(K2), then there exist a Hilbert space h and ν ∈
B(K2,K1 ⊗ h) such that (3.1) is true and
{(φ|⊗1h ν ψ : φ ∈ K1, ψ ∈ K2} = h. (3.2)
2) If in addition to the h′ and ν′ also satisfy the above properties, then there exists a
unique unitary operator U from h to h′ such that ν′ = 1K1 ⊗ U ν.
The right hand side of (3.1) is called a Stinespring dilation of a c.p. map Ξ. If the
condition (3.2) holds, then it is called a minimal.
REDUCED AND EXTENDED WEAK COUPLING LIMIT 11
Remark 3.2. If we choose a basis in h, so that we identify h with Cn, then we can
identify ν with ν1, . . . , νn ∈ B(K2,K1). Then we can rewrite (3.1) as
Ξ(A) =
ν∗jAνj . (3.3)
In the literature, (3.3) is called a Kraus decomposition, even though the work of Stine-
spring is much earlier than that of Kraus.
Note that physically the space h can be interpreted as a part of the reservoir that
directly interacts with the small system.
Proof of Theorem 3.1. Let us prove 1). We equip the algebraic tensor product H0 :=
B(K1)⊗K2 with the following scalar product: for
Xi ⊗ vi, w̃ =
Yi ⊗ wi ∈ H0
we set
(ṽ|w̃) =
(vi|Ξ(X∗i Yj)wj).
By the complete positivity, it is positive definite.
Next we note that there exists a unique linear map π0 : B(K1) → B(H0) satisfying
π0(A)ṽ :=
AXi ⊗ vi.
We check that
(π0(A)ṽ|π0(A)ṽ) ≤ ‖A‖2(ṽ|ṽ), π0(AB) = π0(A)π0(B), π0(A∗) = π0(A)∗.
Let N be the set of ṽ ∈ H0 with (ṽ|ṽ) = 0. Then the completion of H := H0/N is a
Hilbert space. There exists a nondegenerate ∗-representation π of B(K1) in B(H) such
π(A)(ṽ +N ) = π0(A)ṽ.
Using the fact that all our spaces are finite dimensional we see that for some Hilbert
space h we can identify H with K1 ⊗ h and π(A) = A⊗ 1.
We set
νv := 1⊗ v +N .
We check that
Ξ(A) = ν∗A⊗1 ν.
This ends the proof of the existence of the Stinespring dilation.
Let us now prove 2). If h′, ν′ is another pair that gives a Stinespring dilation, we
check that
Xi ⊗ 1h ν vi
Xi ⊗ 1h′ ν′ vi
Therefore, there exists a unitary U0 : K2 ⊗ h → K2 ⊗ h′ such that U0ν = ν′. We check
that U0 A ⊗ 1h = A ⊗ 1h′ U0. Therefore, there exists a unitary U : h → h′ such that
U0 = 1⊗ U .
12 J. DEREZIŃSKI AND W. DE ROECK
We will need the following inequality for c.p. maps:
Theorem 3.3 (Kadison-Schwarz inequality for c.p. maps.). If Ξ is 2-positive
and Ξ(1) is invertible, then
Ξ(A)∗Ξ(1)−1Ξ(A) ≤ Ξ(A∗A). (3.4)
Proof. Let z ∈ C.
A∗A zA∗
z̄A |z|2
≥ 0 implies
Ξ(A∗A) zΞ(A∗)
z̄Ξ(A) |z|2Ξ(1)
≥ 0. Hence, for
φ, ψ ∈ K,
(φ|Ξ(A∗A)φ) + 2Rez̄(ψ|Ξ(1)−1/2Ξ(A)φ) + |z|2(ψ|ψ) ≥ 0. (3.5)
Therefore,
(φ|Ξ(A∗A)|φ)(ψ|ψ) ≥ |(ψ|Ξ(1)−1/2Ξ(A)φ)|2, (3.6)
which implies (3.4).
3.2. Completely positive semigroups. Let K be a finite dimensional Hilbert space. Let
us consider a c.p. semigroup on B(K). We will always assume the semigroup to be
continuous, so that it can be written as etM for a bounded operatorM on B(K). We will
call etM Markov if it preserves the identity.
C.p. Markov semigroups appear in the literature under various names. Among them
let us mention quantum Markov semigroups and quantum dynamical semigroups.
IfM1,M2 are the generators of (Markov) c.p. semigroups and c1, c2 ≥ 0, then c1M1+
c2M2 is the generator of a (Markov) c.p. semigroup. This follows by the Trotter formula.
Here are two classes of examples of c.p. semigroups:
1) Let Υ = Θ+ i∆ be an operator on K, with Θ,∆ self-adjoint. Then
M(A) := iΥA− iAΥ∗ = i[Θ, A]− [∆, A]+
is the generator of a c.p. semigroup and
etM (A) = eitΥAe−itΥ
2) Let Ξ be a c.p. map on B(K). Then it is the generator of a c.p. semigroup and
etΞ(A) =
Ξj(A).
Let Θ, ∆ be self-adjoint operators on K. Let h be an auxiliary Hilbert space and
ν ∈ B(K,K ⊗ h). Then it follows from what we wrote above that
M(S) = i[Θ, A]− [∆, A]+ + ν∗ A⊗1 ν, A ∈ B(K), (3.7)
is the generator of a c.p. semigroup. etM is Markov iff 2∆ = ν∗ν.
The following theorem gives a complete characterization of generators of c.p. semi-
groups on a finite dimensional space [Li, GKS].
Theorem 3.4 (Lindblad, Gorini-Kossakowski-Sudarshan). 1) Let etM be a
c.p. semigroup on B(K) for a finite dimensional Hilbert space K. Then there exist
self-adjoint operators Θ, ∆ on K, an auxiliary Hilbert space h and an operator
ν ∈ B(K,K ⊗ h) such that M can be written in the form (3.7) and
{(φ|⊗1 ν ψ : φ, ψ ∈ K} = h. (3.8)
REDUCED AND EXTENDED WEAK COUPLING LIMIT 13
2) We can always choose Θ and ν so that
TrΘ = 0, Tr ν = 0.
(Above, we take the trace of ν on the space K obtaining a vector in h). If this is the
case, then Θ and ∆ are determined uniquely, and ν is determined uniquely up to
the unitary equivalence.
We will say that a c.p. semigroup is purely dissipative if Θ = 0. We will call (3.7) a
Lindblad form of M . We will say that it is minimal iff (3.8) holds.
Remark 3.5. If we identify h with Cn, then we can write
ν∗ A⊗1 ν =
ν∗jAνj .
Then Tr ν = 0 means Tr νj = 0, j = 1, . . . , n.
Proof of Theorem 3.4. Let us prove 1). The unitary group onK, denoted U(K), is compact.
Therefore, there exists the Haar measure on U(K), which we denote dU . Note that
UXU∗dU = TrX.
Define
iΘ−∆0 :=
M(U∗)UdU,
where Θ and ∆0 are self-adjoint.
Let us show that
M(XU∗)UdU = (iΘ−∆0)X. (3.9)
First check this identity for unitary X , which follows by the invariance of the measure
dU . But every operator is a linear combination of unitaries. So (3.9) follows in general.
We can apply the Kadison-Schwarz inequality to the semigroup etM :
etM (X)∗etM (1)−1etM (X) ≤ etM (X∗X). (3.10)
Differentiating (3.10) at t = 0 yields
M(X∗X) +X∗M(1)X −M(X∗)X −X∗M(X) ≥ 0. (3.11)
Replacing X with UX , where U is unitary, we obtain
M(X∗X) +X∗U∗M(1)UX −M(X∗U∗)UX −X∗U∗M(UX) ≥ 0. (3.12)
Integrating (3.12) over U(K) we get
M(X∗X) +X∗X TrM(1)− (iΘ−∆0)X∗X −X∗X(−iΘ−∆0)∗ ≥ 0. (3.13)
Define
∆1 := ∆0 +
TrM(1),
Ξ(A) := M(A)− (iΘ −∆1)A−A(−iΘ −∆1).
Using (3.13) we see that Ξ is positive. A straightforward extension of the above argument
shows that Ξ is also completely positive. Hence, by Theorem 3.1 1), it can be written as
Ξ(A) = ν∗1 A⊗1 ν1,
14 J. DEREZIŃSKI AND W. DE ROECK
for some auxiliary Hilbert space h and a map ν1 : K → K⊗h.
Finally, let us prove 2). The operator Θ has trace zero, because
i TrΘ− Tr∆0 =
U1M(U
∗)UU∗1 dUdU1
U2UM(U
∗)U∗2 dUdU2
= −i TrΘ− Tr∆0.
Let w be an arbitrary vector in h and
∆ := ∆1 + ν
∗1⊗|w) + 1
(w|w),
ν := ν1 + 1⊗|w).
Then the same generator of a c.p. semigroup can be written in two Lindblad forms:
(iΘ−∆1)A+A(−iΘ−∆1) + ν∗1Aν1,
= (iΘ−∆)A+A(−iΘ −∆) + ν∗Aν.
In particular, choosing w := −Tr ν1, we can make sure that Tr ν = 0.
3.3. Classical Markov semigroups. It is instructive to compare c.p. Markov semigroups
with usual (classical) Markov semigroups.
Consider the space Cn. For u = (u1, . . . , un) ∈ Cn we will write u ≥ 0 iff u1, . . . , un ≥
0. We define 1 := (1, . . . , 1). We say that a linear map T is pointwise positive iff u ≥ 0
implies Tu ≥ 0. We say that it is Markov iff T1 = 1.
A one-parameter semigroup Rt 7→ Tt ∈ B(Cn) will be called a (classical) Markov
semigroup iff Tt is pointwise positive and Markov for any t ≥ 0.
Every continuous one-parameter semigroup on Cn is of the form R+ ∋ t 7→ etm for
some n × n matrix m. Clearly, the transformations etm are pointwise positive for any
t ≥ 0 iff mij ≥ 0, i 6= j. They are Markov for any t ≥ 0 iff in addition
jmij = 0.
Markov c.p. semigroups often lead to classical Markov semigroups, as described in
the following easy fact:
Theorem 3.6. Let P1, . . . , Pn ∈ B(K) satisfy P ∗j = Pj and PjPk = δjkPj. Let P be the
(commutative) ∗-algebra generated by P1, . . . , Pn. Clearly, P is naturally isomorphic to
n. Let etM be a Markov c.p. semigroup on B(K) that preserves the algebra P. Then
is a classical Markov semigroup.
Conversely, from a classical Markov semigroup one can construct c.p. Markov semi-
groups:
Theorem 3.7. Let etm be a classical Markov semigroup on Cn. Let e1, . . . , en denote
the canonical basis of Cn and Eij := |ei)(ej |. Let θ1, . . . θn be real numbers and set
Θ := θ1E11 + · · ·+ θnEnn. For A ∈ B(Cn) define
M(A) := i[Θ, A]− 1
mjj [Ejj , A]+ +
mijEijAEji. (3.14)
Then M is the generator of a Markov c.p. semigroup on B(Cn). The algebra P gener-
ated by E11, . . . , Enn is preserved by e
tM and naturally isomorphic to Cn. Under this
REDUCED AND EXTENDED WEAK COUPLING LIMIT 15
identification, M
equals m.
3.4. Invariant c.p semigroups. Let K be a self-adjoint operator on K. Let M be the
generator of a c.p. semigroup on K. We say that M is K-invariant iff
M(A) = e−itKM(eitKAe−itK)eitK , t ∈ R. (3.15)
We will see later on that c.p. semigroups obtained in the weak coupling limit are always
K-invariant with respect the Hamiltonian of the small system.
Note that M can be split in a canonical way into M = i[Θ, ·] +Md, where Md is its
purely dissipative part. M is K-invariant iff [Θ,K] = 0 and Md is K-invariant. Thus in
what follows it is enough to restrict ourselves to the purely dissipative case.
The following two theorems extend Theorem 3.6 and 3.7.
Theorem 3.8. Consider the set-up of Theorem 3.6. Suppose in addition that K is a
self-adjoint operator on K with the eigenvalues k1, . . . , kn and Pj = 1kj (K). Let M
be K-invariant. Then the algebra P is preserved by etM (and hence the conclusion of
Theorem 3.6 holds).
Theorem 3.9. Consider the set-up of Theorem 3.7. If k1, . . . , kn are real and K :=
k1E11 + · · ·+ knEnn, then M is K-invariant.
The following theorem describes the K-invariance on the level of a Lindblad form.
We restrict ourselves to the Markov case.
Theorem 3.10. Let ν ∈ B(K,K⊗h) and let Y be a self-adjoint operator on h such that
M(A) = −1
[ν∗ν,A]+ + ν
∗A⊗ 1ν, (3.16)
νK = (K⊗1 + 1⊗Y )ν. (3.17)
Then M is the generator of a K-invariant purely dissipative Markov c.p. semigroup.
Proof. We check that ν∗ν commutes with K. Then it is enough to verify that A 7→
ν∗A⊗ 1 ν is K-invariant.
There exists a partial converse of Theorem 3.10.
Theorem 3.11. Let M be the generator of a K-invariant purely dissipative Markov c.p.
semigroup. Let h, ν realize its minimal Lindblad form (3.16). Then there exists a self-
adjoint operator Y on h such that (3.17) is true.
Proof. By the uniqueness part of Theorem 3.1 there exists a unique unitary operator Ut
on h such that eitK⊗Ut ν e−itK = ν. We easily check the Ut is a continuous 1-parameter
unitary group so that Ut can be written as e
itY for some self-adjoint Y .
Note that Theorems 3.10 and 3.11 have a clear physical meaning. The operator ν is
responsible for “quantum jumps”. The operator Y describes the energy of the reservoir
(or actually of the part of the reservoir “directly seen” by the interaction). The equation
(3.17) describes the energetic balance in each quantum jump.
3.5. Detailed Balance Condition. In the literature the name Detailed Balance Condition
(DBC) is given to several related but non-equivalent concepts. In this subsection we
discuss some of the versions of the DBC relevant in the weak coupling limit.
16 J. DEREZIŃSKI AND W. DE ROECK
Some of the definitions of the DBC (both for classical and quantum systems) involve
the time reversal [Ag, Ma, MaSt]. In the weak coupling limit one does not need to
introduce the time reversal, hence we will only discuss versions of the DBC that do not
involve this operation. (See however [DM] for a discussion of time-reversal in semigroups
obtained in the weak coupling limit.)
Let us first recall the definition of the classical Detailed Balance Condition. Let p =
(p1, . . . , pn) ∈ Cn be a vector with p1, . . . , pn > 0. Introduce the scalar product on Cn:
(u|u′)p :=
jpj . (3.18)
Let etm be a classical Markov semigroup on Cn. We say that m satisfies the Detailed
Balance Condition for p iff m is self-adjoint for (·|·)p.
Let us now consider the quantum case. Let ρ be a nondegenerate density matrix. As
usual, we assume that K is finite dimensional. On B(K) we introduce the scalar product
(A|B)ρ := Tr ρ1/2A∗ρ1/2B. (3.19)
Let M be the generator of a c.p. semigroup on B(K). Recall that it can be uniquely
represented as
M = i[Θ, ·] +Md,
where Md is its purely dissipative part and i[Θ, ·] its Hamiltonian part. We say that M
satisfies the Detailed Balance Condition (or DBC) for ρ iff Md is self-adjoint and i[Θ, ·]
is anti-self-adjoint for (·|·)ρ.
Note that M satisfies the DBC for ρ iff [Θ, ρ] = 0 and Md satisfies the DBC for ρ.
Therefore, in our further analysis we will often restrict ourselves to the purely dissipative
case.
We believe that in the quantum finite dimensional case the above definition of the
DBC is the most natural. It was used e.g. in [DF1] under the name of the standard
Detailed Balance Condition.
A similar but different definition of the DBC can be found in [FKGV, Al1]. Its only
difference is the replacement of the scalar product (·|·)ρ given in (3.19) with
Tr ρA∗B. (3.20)
Note that if M is K-invariant and ρ is a function of K, then both definitions are equiv-
alent.
The weak coupling limit applied to a small system with a Hamiltonian K interact-
ing with a thermal reservoir at some fixed temperature β always yields a Markov c.p.
semigroup that is K-invariant and satisfies the DBC for ρ = e−βK/Tr eβK ; see e.g.
[LeSp, DF1] and Subsect 4.3.
There exists a close relationship between the classical and quantum DBC.
Theorem 3.12. Consider the set-up of Theorem 3.6. Let ρ be a density matrix on K with
the eigenvalues p1, . . . , pn and let Pj equal the spectral projections of ρ for the eigenvalue
pj. If M satisfies the DBC for ρ, either in the sense of (3.19) or in the sense of (3.20),
then the classical Markov semigroup etM
satisfies the DBC for p = (p1, . . . , pn).
REDUCED AND EXTENDED WEAK COUPLING LIMIT 17
Theorem 3.13. Consider the set-up of Theorem 3.7. Let etm satisfy the classical DBC
for p = (p1, . . . , pn). Then M defined by (3.14) satisfies both quantum versions of the
DBC for ρ := p1E11 + · · ·+ pnEnn.
The following theorem describes the DBC for K-invariant generators on the level of
their Lindblad form. It is an extension of Theorem 3.10. (Note that (3.21), (3.22) are
identical to (3.16), (3.17) of Theorem 3.10).
Theorem 3.14. Let ν ∈ B(K,K⊗h) and Y a self-adjoint operator on h such that
M(A) = −1
[ν∗ν,A]+ + ν
∗A⊗ 1ν, (3.21)
νK = (K⊗1 + 1⊗Y ) ν, (3.22)
Trh νAν
∗ = ν∗ A⊗e−βY ν. (3.23)
Then M is the generator of a K-invariant purely dissipative Markov c.p. semigroup
satisfying the DBC for ρ := e−βK/Tr e−βK.
Proof. It follows from (3.17) that ν∗ν commutes with e−βK/2. Hence [ν∗ν, ·]+ is self-
adjoint for (·|·)ρ.
If M is a map on B(K), then M∗ρ will denote the adjoint for this scalar product. Let
M1(A) = ν
∗ A⊗1 ν. We compute:
1 (A) = Trh e
βK/2⊗1 ν e−βK/2Ae−βK/2 ν∗ eβK/2⊗1
= Trh e
βK/2ν∗ (e−βK/2Ae−βK/2⊗e−βY ) ν eβK/2 (3.24)
= ν∗ A⊗1ν =M1(A). (3.25)
In (3.24) and (3.25) we used (3.23) and (3.22) respectively.
It is possible to replace the condition (3.23) with a different condition (3.26). Note
that whereas (3.23) is quadratic in ν, (3.26) is linear in ν.
Theorem 3.15. Suppose that ǫ is an antiunitary operator on h such that
(φ⊗w|νψ) = (νφ|ψ ⊗ e−βY/2ǫw), φ, ψ ∈ K, w ∈ h. (3.26)
Then (3.23) holds.
Proof. It is sufficient to assume that A = |ψ)(ψ| for some ψ ∈ K. Let φ ∈ K. Let
{wi | i ∈ I} be an orthonormal basis in h. Then
Trh(φ|νAν∗φ) =
(φ⊗wi|νψ)(νψ|φ⊗wi)
(νφ|ψ⊗e−βY/2ǫwi)(ψ⊗e−βY/2ǫwi|νφ)
|ψ)(ψ|⊗e−βY νφ
= (φ|ν∗ A⊗e−βY νφ).
There exists an extension of Theorem 3.11 to the Detailed Balance Condition. It can
be viewed as a partial converse of Theorems 3.14 and 3.15:
18 J. DEREZIŃSKI AND W. DE ROECK
Theorem 3.16. Let M be the generator of a K-invariant purely dissipative Markov c.p.
semigroup satisfying the DBC for e−βK/Tr e−βK . Let h, ν realize its minimal Lindblad
form (3.21). Let a self-adjoint operator Y on h satisfy (3.22). Then (3.23) is true and
there exists a unique antiunitary operator ǫ on h such that (3.26) holds. Besides, ǫY ǫ =
−Y and ǫ2 = 1.
Proof. Step 1. By the proof of Theorem 3.14, the DBC for e−βK/Tr e−βK together with
(3.22) imply (3.23).
Step 2. The next step is to prove that (3.23) and (3.8) imply the existence of an antiu-
nitary ǫ on h satisfying (3.26).
Identify h with Cn, so that we have a complex conjugation w 7→ w in h. We can
assume that Y is diagonal, so that Y w = Y w, w ∈ h. Define ν⋆ by
(φ⊗w|νψ) = (ν⋆φ|ψ ⊗ w̄), φ, ψ ∈ K, w ∈ h. (3.27)
(Note that ⋆ is a different star from ∗ denoting the Hermitian conjugation, see [DF1]).
We can rewrite (3.23) as
ν⋆∗ A⊗1 ν⋆ = ν∗ 1⊗e−βY/2 (A⊗1) 1⊗e−βY/2 ν. (3.28)
(3.28) defines a c.p. map. By the uniqueness part of Theorem 3.1 and (3.8), we
obtain the existence of a unitary map U on h such that ν⋆ = 1⊗Ue−βY/2 ν. Now we set
ǫw = U∗w̄.
Step 3. We apply (3.23) twice:
(φ⊗w|νψ) = (νφ|ψ⊗e−βY/2ǫw) = (φ⊗(e−βY/2ǫ)2w|νψ).
Using (3.8) we obtain w = (e−βY/2ǫ)2w.
Step 4. Finally applying (3.23) together with (3.22) twice we obtain
(φ⊗w|νψ) = (νeβK/2φ|e−βK/2ψ⊗ǫw) = (φ⊗ǫ2w|νψ).
Thus with help of (3.8) we get w = ǫ2w.
Note that the above results show that for c.p. Markov semigroups that areK-invariant
and satisfy the DBC for e−βK/Tr e−βK we naturally obtain a certain algebraic structure
on the “restricted reservoir” h that resembles closely the famous Tomita-Takesaki theory.
The properties of e−βY and ǫ are paralel to those of the modular operator and the modular
conjugations – the basic objects of the Tomita-Takesaki formalism. (See also Subsection
4.3).
4. Bosonic reservoirs. In this section we recall basic terminology related to second
quantization, see e.g. [De0]. We also introduce Pauli-Fierz operators – a class of models
(known in the literature under various names) that are often used to describe realistic
physical systems, see e.g. [DJ1, DJP].
4.1. Second quantization. Let HR be a Hilbert space describing 1-particle states. The
corresponding bosonic Fock space is defined as
Γs(HR) :=
⊗ns HR.
REDUCED AND EXTENDED WEAK COUPLING LIMIT 19
The vacuum vector is Ω = 1 ∈ ⊗0sHR = C.
If z ∈ HR, then
a(z)Ψ :=
n(z|⊗1(n−1)⊗Ψ ∈ ⊗n−1s HR, Ψ ∈ ⊗ns HR,
is called the annihilation operator of z and a∗(z) := a(z)∗ is the corresponding creation
operator. They are closable operators on Γs(HR).
For an operator q on HR we define the operator Γ(q) on Γs(HR) by
= q ⊗ · · · ⊗ q. (4.1)
For an operator h on HR we define the operator dΓ(h) on Γs(HR) by
dΓ(h)
= h⊗ 1(n−1)⊗ + · · · 1(n−1)⊗ ⊗ h.
Note the identity Γ(eith) = eitdΓ(h).
4.2. Coupling to a bosonic reservoir. Let K be a finite dimensional Hilbert space. We
imagine that it describes a small quantum system interacting with a bosonic reservoir
described by the Fock space Γs(HR). The coupled system is described by the Hilbert
space H := K⊗ Γs(HR).
Let V ∈ B(K,K ⊗HR). For Ψ ∈ K ⊗⊗ns HR we set
a(V )Ψ :=
nV ∗⊗1(n−1)⊗Ψ ∈ K ⊗⊗n−1s HR.
a(V ) is called the annihilation operator of V and a∗(V ) := a(V )∗ the corresponding
creation operator. They are closable operators on K⊗Γs(HR). Note in particular that if
V is written in the form
j Vj⊗|bj) (which is always possible), then
a∗(V ) =
Vj ⊗ a∗(bj), a(V ) = V ∗j ⊗ a(bj),
where a∗(bj), a(bj) are the usual creation/annihilation operators introduced in the pre-
vious subsection.
The following class of operators plays the central role in our article:
Hλ = K ⊗ 1 + 1⊗ dΓ(HR) + λ(a∗(V ) + a(V )). (4.2)
HereK is a self-adjoint operator describing the free dynamics of the small system, dΓ(HR)
describes the free dynamics of the reservoir and a∗(V )/a(V ), for some V ∈ B(K,K⊗HR),
describe the interaction. Operators of the form 4.2 will be called Pauli-Fierz operators.
Note that operators of the form (4.2) or similar are very common in the physics lit-
erature and are believed to give an approximate description of realistic physical systems
in many circumstances (e.g. an atom interacting with radiation in the dipole approxima-
tion), see e.g. [DJ1].
4.3. Thermal reservoirs. In this subsection we will discuss thermal reservoirs. We fix a
positive number β having the interpretation of the inverse temperature.
20 J. DEREZIŃSKI AND W. DE ROECK
Recall that the free Hamiltonian is H0 := K⊗1 + 1⊗dΓ(HR). To have a simpler
formula for the Gibbs state of the small system we assume that Tr e−βK = 1. We set
τt(C) := e
itH0Ce−itH0 ,
ωβ(C) := Tr e
−βK⊗|Ω)(Ω| C, C ∈ B(H).
Theorem 4.1. The following are equivalent:
1) For any D1, D2, D
2 ∈ B(K) and
Bj := Dj⊗1 (a∗(V ) + a(V )) D′j ⊗ 1, j = 1, 2,
and for any t ∈ R we have
ωβ(τt(B1)B2) = ωβ (B2τt+iβ(B1)) . (4.3)
2) For any function f on the spectrum of spHR and A ∈ B(K), we have
TrHR 1⊗f̄(−HR) V A V ∗ = V ∗ A⊗e−βHRf(HR) V. (4.4)
Proof. The left hand side of (4.3) equals
Tr e−βK+itKD1V
∗(D′1e
−itKD2⊗e−itHR)V D′2.
The right hand side of (4.3) equals
TrD2V
∗(D′2e
−βK+itKD1⊗e(−β+it)HR)V D′1e−itK .
Now we set A1 := D
−βK+itKD1, A2 := D
−itKD2, and use the cyclicity of the trace.
We obtain
TrA2⊗e−itHR V A1 V ∗ = TrA2V ∗A1⊗e−βHR+itHR V.
By the Fourier transformation we get
TrA2⊗f̄(−HR) V A1 V ∗ = TrA2V ∗A1⊗e−βHRf(HR) V.
This implies (4.4).
We will say that the reservoir is thermal at the inverse temperature β iff the conditions
of Theorem 4.1 are true.
(4.3) is just the β-KMS condition for the state ωβ , the dynamics τ and appropriate
operators. Note that (4.3) is satisfied for Pauli-Fierz semi-Liouvilleans constructed with
help of the Araki-Woods representations of the CCR, where we use the terminology of
[DJP, De0]. Theorem 4.1 describes a substitute of the KMS condition without invoking
explicitly operator algebras.
The KMS condition is closely related to the Tomita-Takesaki theory. One of the
objects introduced in this theory is the modular conjugation. It turns out that the set-up
of Theorem 4.1 is sufficient to introduce a substitute for the modular conjugation without
talking about operator algebras.
Define
HR̃ := {(φ|⊗f(HR) V ψ : φ, ψ ∈ K, f ∈ Cc(R)}
(cl denotes the closure). Clearly, HR̃ is a subspace of HR invariant with respect to the
1-particle reservoir Liouvillean HR. It describes the part of HR that is coupled to the
small system. Let HR̃ denote the operator HR restricted to the space HR̃.
REDUCED AND EXTENDED WEAK COUPLING LIMIT 21
Theorem 4.2. Suppose that the reservoir is thermal at inverse temperature β. Then
there exist a unique antiunitary operator ǫR̃ on HR̃ such that
(φ⊗w|V ψ) = (V φ|ψ⊗e−βHR̃ǫR̃w). (4.5)
It satisfies ǫ2
= 1 and ǫR̃HR̃ǫR̃ = −HR̃.
Proof. For f ∈ Cc(R), φ, ψ ∈ K, we set
(φ|⊗e−βHR̃/2f(HR̃) V ψ
:= (ψ|⊗f̄(−HR̃)V φ.
(4.4) implies that ǫR̃ is a well defined antiunitary map.
5. Quantum Langevin dynamics. Suppose that we are given a c.p. Markov semigroup
etM on B(K). We will describe a certain class of self-adjoint operators Z on a larger
Hilbert space such that e−itZ · eitZ is a dilation on etM . We will use the name quantum
Langevin (or stochastic) dynamics for e−itZ · eitZ . The unitary group e−itZ will be called
a Langevin (or stochastic) Schrödinger dynamics.
In Subsection 5.1 we will restrict ourselves to a subclass of quantum Langevin dy-
namics involving only the so-called linear noises. Actually, at present our results on the
extended weak coupling limit are limited only to them.
In Subsection 5.2 we will describe a more general class of quantum Langevin dynamics,
which also involve quadratic noises. Our construction involving quadratic noises is related
to the operator-theoretic approach of Chebotarev [Ch, ChR], and especially of Gregoratti
[Gr].
We expect that our approach to the extended weak coupling limit can be improved
to cover also this larger class. Within the approach of [AFL] there exist partial results in
this direction [Go].
The history of the discovery of quantum Langevin dynamics is quite involved. The
construction can be traced back to [AFLe], and especially [HP] where the quantum
stochastic calculus was introduced. But apparently only in [Fr] and [Maa] it was inde-
pendently realized that this leads to a dilation of Markov c.p. semigroups. Let us also
mention [At, Me, Fa] for more recent presentations of the quantum stochastic calculus.
5.1. Linear noises. Apart from a c.p. Markov semigroup etM let us fix some additional
data. More precisely, we fix an operator Υ, an auxiliary Hilbert space h and an operator
ν from K to K ⊗ h such that
−iΥ + iΥ∗ = −ν∗ν
and M is given by
M(A) = −i(ΥA−AΥ∗) + ν∗ A⊗1 ν, A ∈ B(K).
In other words, we fix a concrete Lindblad form of M .
Introduce the Hilbert space ZR := h ⊗ L2(R). The enlarged Hilbert space is Z :=
K ⊗ Γs(ZR).
Let ZR be the operator of multiplication by the variable x on L
2(R). Let (1|, |1) be
defined as in (2.2).
22 J. DEREZIŃSKI AND W. DE ROECK
We choose a basis (bj) in h, so that we can write
νj ⊗ |bj). (5.1)
(Note that at the end the construction will not depend on the choice of a basis). Set
ν+j = νj , ν
j = ν
For t ≥ 0 we define the quadratic form
Ut := e
−itdΓ(ZR)
t≥tn≥···≥t1≥0
dtn · · · dt1
×(2π)−n2
j1,...,jn
ǫ1,...,ǫn∈{+,−}
×(−i)ne−i(t−tn)Υνǫnjn e
−i(tn−tn−1)Υ · · · νǫ1j1 e
−i(t1−0)Υ
k=1,...,n: ǫk=+
a∗(eitkZRbjk ⊗ |1))
k′=1,...,n: ǫk′=−
a(eitk′ZRbjk′ ⊗ |1));
U−t := U
We will denote by IK the embedding of K ≃ K ⊗ Ω in Z.
Theorem 5.1. Ut extends to a strongly continuous unitary group on Z such that
I∗KUtIK = e
−itΥ,
I∗KUt A⊗ 1 U−tIK = etM (A).
Thus Ut is a unitary dilation of e
−itΥ, and Ut · U∗t is a dilation of etM .
As every strongly continuous unitary group, Ut can be written as e
−itZ for a certain
self-adjoint operator Z. Note that formally (and also rigorously with an appropriate
regularization)
(Υ + Υ∗) + dΓ(ZR)
+(2π)−
2 a∗ (ν ⊗ |1)) + (2π)− 12 a (ν ⊗ |1)) .
Thus Z has the form of a Pauli-Fierz operator with a rather singular interaction.
Let us present an alternative variation of the above construction, which is actually
closer to what can be found in the literature. Let F be the Fourier transformation on
ZR = h ⊗ L2(R) defined as in (2.6). The operator Z transformed by 1K⊗Γ(F) will be
denoted by
Ẑ := 1K⊗Γ(F) Z 1K⊗Γ(F∗). (5.2)
It equals
(Υ + Υ∗)⊗1 + 1⊗dΓ(Dτ )
+a (ν ⊗ |δ0)) + a∗ (ν ⊗ |δ0)) ,
REDUCED AND EXTENDED WEAK COUPLING LIMIT 23
where δ0, Dτ are defined as in (2.8), (2.9).
Similarly to the operator of Section 2.1 denoted with the same symbol, the operator
Ẑ (as well as Z) has a number of intriguing properties. Let us describe one of them.
Let D0 := h ⊗ H1(R). (Recall that H1(R) is the first Sobolev space). Let
Γs(D0),
denote the corresponding algebraic Fock space and D1 := K ⊗
Γs(D0). Introduce the
(non-self-adjoint) sesquilinear form
Ẑ+ = Υ⊗1 + 1⊗dΓ(Dτ ) (5.3)
+a (ν ⊗ |δ0)) + a∗ (ν ⊗ |δ0)) .
Let ψ, ψ′ ∈ D1. Then
(ψ|(e−itẐ − 1)ψ′) = −i(ψ|Ẑ+ψ′). (5.4)
Thus it seems that Ẑ+ = Ẑ, which is true only if Υ is self-adjoint and hence there are
no off-diagonal terms in Z. Clearly, the explanation of the above paradox is similar as
in Subsect. 2.1: (ψ|e−itẐψ′) is not differentiable at zero. This is related to the fact that
ψ, ψ′ do not belong to DomZ. Thus Ẑ+ can again be called a false form.
In the literature, the Langevin Schrödinger dynamics e−itẐ is usually introduced
through the so-called Langevin (or stochastic) Schrödinger equation satisfied by
Ŵ (t) := eitdΓ(Dτ )e−itẐ . (5.5)
To write this equation recall the decomposition (5.1) and note that. Then, in the sense
of quadratic forms on D1, we have
Ŵ (t) =
Υ⊗1 + a∗ (ν ⊗ |δt))
Ŵ (t) +
ν∗j Ŵ (t)a (bj ⊗ |δt)) . (5.6)
Note that a (ν ⊗ |δ0)) and a∗ (ν ⊗ |δ0)) appearing in Ẑ and Ẑ+ are quantum analogs
of a classical white noise. They are “localized” at τ = 0. Besides, they are (formally)
given by a linear expression in terms of creation/annihilation operators. Therefore, they
are often called linear quantum noises.
5.2. Quadratic noises. This subsection is outside of the main line of this article. It is
closely related to Subsect. 2.2. It is not needed for the description of the weak coupling
limit, as given in the next section.
Clearly, Ψ ∈ K⊗
⊗ns h⊗ L2(R)
⊗ns L2(R, h)
can be identified with a function
Ψ(τ1, . . . , τn) with values in K⊗ (⊗nh) and the arguments satisfying τ1 < · · · < τn.
Let S be a unitary operator on K ⊗ h. Let S(j) be this operator acting on K ⊗ ⊗nh,
where it is applied to the j’th “leg” of the tensor product ⊗nh. We define an operator
Λ(S) on K ⊗
⊗ns L2(R, h)
as follows: If τ1 < · · · < τk < 0 < τk+1 < · · · < τn, then
(Λ(S)Ψ) (τ1, . . . , τn) := S(k+1) · · ·S(n)Ψ(τ1, . . . , τn).
Clearly, Λ(S) is a unitary operator. If K = C, then it coincides with Γ(γ(S)), where γ(S)
was defined in (2.15) and Γ is the functor of the second quantization defined in (4.1).
24 J. DEREZIŃSKI AND W. DE ROECK
Introduce the operator ẐS,0 on K⊗Γs(ZR) by
ẐS,0 := Υ + Λ(S)
∗1⊗dΓ(Dτ )Λ(S). (5.7)
The operator (5.7) is very singular and contains a “delta interaction at τ = 0”.
Let us now define the dynamics ÛS,t that generalizes Ût. Let Sij ∈ B(K) be defined
Sij⊗|bi)(bj |. (5.8)
ν+S,j = νj , ν
S,j =
ν∗i Sij .
Then we introduce the quadratic form
ÛS,t :=
t≥tn≥···≥t1≥0
dtn · · · dt1
j1,...,jn
ǫ1,...,ǫn∈{+,−}
×(−i)n
k=1,...,n: ǫk=+
a∗(bjk ⊗ |δtk−t))
e−i(t−tn)ẐS,0νǫnS,jn⊗1e
−i(tn−tn−1)ẐS,0 · · · νǫ1S,j1⊗1e
−i(t1−0)ẐS,0
k′=1,...,n: ǫk′=−
a(bjk′ ⊗ |δtk′ ));
ÛS,−t := Û
One can check that ÛS,t extends to a strongly continuous unitary group. Therefore, one
can define a self-adjoint operator ẐS such that ÛS,t = e
−itẐS . It satisfies
I∗KÛS,tIK = e
−itΥ,
I∗KÛS,t A⊗ 1 ÛS,−tIK = etM (A).
It is awkward to write a formula for ẐS in terms of creation/annihilation operators,
even formally. There exists however and alternative formalism that is commonly used in
the literature to define the group e−itẐS . Let ψ, ψ′ ∈ D1. Introduce the cocycle
ŴS(t) := e
itdΓ(Dτ )e−itẐS . (5.9)
Then, in the sense of a quadratic form on D1, the cocycle satisfies the differential equation
ŴS(t) =
Υ⊗1 + a∗(ν⊗|δt))
ŴS(t) (5.10)
i(1 − Sij)⊗a∗(bi ⊗ |δt)) ŴS(t) a(bj⊗|δt)) (5.11)
ν−S,j ŴS(t)a(bj⊗|δt)). (5.12)
REDUCED AND EXTENDED WEAK COUPLING LIMIT 25
This formula is the quantum Langevin (stochastic) equation for the cocycle ŴS(t) in the
sense of [HP, Fa, Pa, At, Maa, Fr, Bar, Me], which includes all three kinds of noises. In
the literature, the dilation e−itẐS is usually introduced through a version of (5.12).
5.3. Total energy operator. Let us analyze the impact of the invariance of a c.p. semigroup
on its quantum Langevin dynamics.
Suppose now that K is a self-adjoint operator on K and Y a self-adjoint operator on
h. Assume that they satisfy
ν K = (K⊗1 + 1⊗Y )ν,
(Υ + Υ∗),K
= 0. (5.13)
This implies in particular that M is K-invariant. Define the self-adjoint operator on Z
E := K⊗1 + 1⊗dΓ(Y⊗1). (5.14)
Then it is easy to see that the quantum Langevin dynamics commutes with this operator:
[E, e−itZ ] = 0. (5.15)
E will be called the total energy operator, which is a name suggested by the physical
interpretation that we attach to E.
Next we discuss the implications of the DBC of a c.p. semigroup on its quantum
Langevin dynamics. We set
σt(C) := e
itECe−itE ,
ωβ(C) := Tr e
−βK⊗|Ω)(Ω| C/Tr e−βK , C ∈ B(Z).
We will see that the DBC for e−βK/Tr e−βK is related to a version of the β-KMS con-
dition for the dynamics σt and the state ωβ .
Theorem 5.2. Assume (5.14). Then the following statements are equivalent:
1) For any D1, D2, D
2 ∈ B(K), f1, f2 ∈ L2(R) and
Bj := Dj⊗1
a∗(ν⊗|fj)) + a(ν⊗|fj))
D′j⊗1, j = 1, 2.
and for any t ∈ R we have
ωβ(σt(B1)B2) = ωβ (B2σt+iβ(B1)) . (5.16)
Trh νAν
∗ = ν∗ A⊗e−βY ν, (5.17)
(This implies in particular that M satisfies the DBC for e−βK/Tr e−βK).
6. Weak coupling limit for Pauli-Fierz operators. In this section we describe the
main results of this article. They are devoted to a rather large class of Pauli-Fierz oper-
ators in the weak coupling limit. In the first subsection we recall the well known results
about the reduced dynamics, which go back to Davies [Da1, Da2, Da3]. In the second
subsection we describe our results that include the reservoir [DD2]. They are inspired by
[AFL]. Finally, we discuss the case of thermal reservoirs.
26 J. DEREZIŃSKI AND W. DE ROECK
6.1. Reduced weak coupling limit. We consider a Pauli-Fierz operator
Hλ = K ⊗ 1 + 1⊗ dΓ(HR) + λ(a∗(V ) + a(V )).
We assume that K is finite dimensional and for any A ∈ B(K) we have
‖V ∗A ⊗
1 e−itH0V ‖dt < ∞. The following theorem is essentially a special case of a result of
Davies [Da1, Da2, Da3], see also [DD2].
Theorem 6.1 (Reduced weak coupling limit for Pauli-Fierz operators). There
exists a K-invariant Markov c.p. semigroup etM on B(K) such that
e−itK/λ
itHλ/λ
A⊗ 1 e−itHλ/λ
itK/λ2 = etM (A),
and a contractive semigroup e−itΥ on K such that [Υ,K] = 0 and
eitK/λ
−itHλ/λ
IK = e
−itΥ.
If the reservoir is at inverse temperature β, then M satisfies the DBC for the state
e−βK/Tr e−βK .
The operator Υ ∈ B(K) arising in the weak coupling limit equals
Υ := −i
k−k′=ω
1k(K)V
∗1k′(K)e
−it(HR−ω)V 1k(K)dt.
In order to write an explicit formula forM it is convenient to introduce an additional
assumption, which anyway will be useful later on in the extended weak coupling limit.
Assumption 6.2. Suppose that for any ω ∈ spK − spK there exist an open Iω ⊂ R and
a Hilbert space hω such that ω ∈ Iω and
Ran1Iω (HR) ≃ hω ⊗ L2(Iω , dx),
1Iω (HR)HR is the multiplication operator by the variable x ∈ Iω and, for ψ ∈ K,
1Iω(HR)V ψ ≃
v(x)ψdx.
Assume that Iω are disjoint for distinct ω and x 7→ v(x) ∈ B(K,K⊗hω) is continuous at
Thus we assume that the reservoir 1-body Hamiltonian HR and the interaction V are
well behaved around the Bohr frequencies – differences of eigenvalues of K.
Let h := ⊕
hω. We define νω ∈ B(K,K⊗hω) by
νω := (2π)
ω=k−k′
1k(K)v(ω)1k′(K)
and ν ∈ B(K,K⊗h) by
REDUCED AND EXTENDED WEAK COUPLING LIMIT 27
Note that
iΥ− iΥ∗ =
k−k′=ω
1k(K)V
∗1k′(K)e
−it(HR−ω)V 1k(K)dt
k−k′=ω
1k(K)v
∗(ω)1k′(K)v(ω) 1k(K)
= ν∗ν.
The generator of a c.p. Markov semigroup that arises in the reduced weak coupling limit,
called sometimes the Davies generator, is
M(A) = −i(ΥA−AΥ∗) + ν∗ A⊗1 ν (6.1)
[A, ν∗ν]+ + ν
∗A⊗1 ν, A ∈ B(K).
6.2. Energy of the reservoir in the weak coupling limit. Introduce the operator Y on h
by setting
Y = ω on hω. (6.2)
The operator Y has the interpretation of the asymptotic energy of the restricted reservoir.
Theorem 6.3. 1) The operator ν constructed in the weak coupling limit satisfies
ν K = (K⊗1 + 1⊗Y )ν. (6.3)
This implies in particular that M is K-invariant.
2) If the reservoir is at inverse temperature β, then ν satisfies
Trh νAν
∗ = ν∗ A⊗e−βY ν, (6.4)
This implies in particular that M satisfies the DBC for e−βK/Tr e−βK .
6.3. Extended weak coupling limit. Recall that given (Υ, ν, h) we can define the space
ZR and the Langevin Schrödinger dynamics e−itZ on the space Z := K ⊗ Γs(ZR), as in
Subsect. 5.1.
For λ > 0, we define the family of partial isometries Jλ,ω : hω⊗L2(R) → hω⊗L2(Iω) ⊂
(Jλ,ωgω)(y) =
), if y ∈ Iω ;
0, if y ∈ R\Iω.
We set Jλ : ZR → HR, defined for g = (gω) by
Jλg :=
Jλ,ωgω.
Note that Jλ are partial isometries and s− limλց0 J∗λJλ = 1.
Set Z0 := dΓ(ZR). The following theorem [DD2] was inspired by [AFL]:
Theorem 6.4 (Extended weak coupling limit for Pauli-Fierz operators).
s∗ − lim
Γ(J∗λ)e
iλ−2tH0e−iλ
−2(t−t0)Hλeiλ
−2t0H0Γ(Jλ)
= eitZ0e−i(t−t0)Ze−it0Z0 .
28 J. DEREZIŃSKI AND W. DE ROECK
The extended weak coupling limit can be used to describe interesting physical prop-
erties of non-equilibrium quantum systems, see e.g. [DM]. The following corollary, which
generalizes the results of [Du], describes the asymptotics of correlation functions for ob-
servables of the form Γ(Jλ)AΓ(J
λ), where A are observables on the asymptotic space.
Corollary 6.5 (Asymptotics of correlation functions). Suppose that
Aℓ, . . . , A1 ∈ B(Z) and t, tℓ, . . . , t1, t0 ∈ R. Then
s∗ − lim
iλ−2tH0e−iλ
−2(t−tℓ)Hλe−iλ
−2tℓH0Γ(Jλ)AℓΓ(J
· · ·Γ(Jλ)A1Γ(J∗λ)eiλ
−2t1H0e−iλ
−2(t1−t0)Hλe−iλ
−2t0H0IK
= I∗Ke
itZ0e−i(t−tℓ)Ze−itℓZ0Aℓ
· · ·A1eit1Z0e−i(t1−t0)Ze−it0Z0IK.
The following corollary is interesting since it describes how reservoir Hamiltonians
converge to operators whose dynamics under the quantum Langevin dynamics U−t · Ut
is well-studied, see e.g. [Bar].
Corollary 6.6 (Asymptotic reservoir energies). Consider the operator Y : h 7→
h defined in (6.2). The operator E := K⊗1 + 1⊗dΓ(Y⊗1) plays the role of “asymptotic
total energy operator”, i.e.
[E, eitZ ] = 0. (6.5)
Besides, for κ1, . . . , κℓ ∈ R,
s∗ − lim
iλ−2tH0e−iλ
−2(t−tℓ)Hλe−iλ
−2tℓH0 eiκℓdΓ(HR)
· · · eiκ1dΓ(HR) eiλ
−2t1H0e−iλ
−2(t1−t0)Hλe−iλ
−2t0H0IK
= I∗Ke
itZ0e−i(t−tℓ)Ze−itℓZ0 eiκℓdΓ(Y⊗1)
· · · eiκ1dΓ(Y⊗1) eit1Z0e−i(t1−t0)Ze−it0Z0IK.
REDUCED AND EXTENDED WEAK COUPLING LIMIT 29
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	Introduction.
	Toy model of the weak coupling limit.
	Dilations of contractive semigroups.
	``Toy quadratic noises''.
	Weak coupling limit for Friedrichs operators.
	Completely positive maps and semigroups.
	Completely positive maps.
	Completely positive semigroups.
	Classical Markov semigroups.
	Invariant c.p semigroups.
	Detailed Balance Condition.
	Bosonic reservoirs.
	Second quantization.
	Coupling to a bosonic reservoir.
	Thermal reservoirs.
	Quantum Langevin dynamics.
	Linear noises.
	Quadratic noises.
	Total energy operator.
	Weak coupling limit for Pauli-Fierz operators.
	Reduced weak coupling limit.
	Energy of the reservoir in the weak coupling limit.
	Extended weak coupling limit.
