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F_~litors
Luigi Accardi
Dipartimento di Matematica, Universit& di Roma II
Via OrazioR aimondo, 00173 Rome, Italy
Wilhelm yon Waldenfels
Institut fL]r Angewandte Mathematik
Universit~t Heidelberg
Im Neuenheimer Feld 294
6900 Heidelberg, Federal Republic ofG ermany
Mathematics Subject Classification (1980): 46LXX, 47DXX, 58Fll, 60FXX,
60GXX, 60HXX, 60JXX, 82A15
ISBN 3-540-53026-6 Springer-Verlag Berlin Heidelberg New York
ISBN 0-38?-53026-6 Springer-Verlag NewYork Berlin Heidelberg
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INTRODUCTION
This volume, the fifth one of the quantum probability series, contains the proceedings
of the Fourth Workshop on Quantum Probability, held in Heidelberg, September 26-30,
.8891
The workshop was made possible by the support of the Deutsche Forschungsgemein-
schaft (Sonderforschungsbereich 123, University of Heidelberg) and of the Centro Vol-
terra, University of Rome II.
We are glad to thank all the participants for their contributions to the present volume
and for their contributions in discussions and conversations.
Luigi Accardi
Wilhelm von Waldenfels
TABLE OF CONTENTS
.L ACCARDI, A. FRIGERIO, L.Y. GANG, Quantum Langevin Equation
in the Weak Coupling Limit
L. ACCARDI, L.Y. GANG, On the Low Density Limit of Boson Models 17
.L ACCARDI, R.L. HUDSON, Quantum Stochastic Flows and Non 54
Abelian Cohomology
.D APPELBAUM, Quantum Diffusions on Involutive Algebras 70
A. BARCHIELLI, Some Markov Semigroups in Quantum Probability 86
.V BELAVKIN, A Quantum Stochastic Calculus in Fock Space of 99
Input and Output Nondemolitlon Processes
C. CECCHINI, .B KOMMERER, Stochastic Transitions on Preduals 126
of yon Neumann Algebras
F. FAGNOLA, Quantum Stochastic Calculus and a Boson Levy 131
Theorem
K.-H. FICHTNER, .U SCHREITER, Locally Independent Boson Systems 145
A. FRIGERIO, Time-Inhomogeneous and Nonlinear Quantum 162
Evolutions
.D GODERIS, A. VERBEURE, .P VETS, Quantum Central Limit and 178
Coarse Graining
G.C. HEGERFELDT, An Open Problem in Quantum Shot Noise 194
.E HENSZ, A Method of Operator Estimation and a Strong Law 204
of Large Numbers in yon Neumann Algebras
A.S. HOLEVO, An Analog of the Ito Decomposition for 211
Multiplicative Processes with Values in a Lie Group
R.L. HUDSON, .P SHEPPERSON, Stochastic Dilations of Quantum 216
Dynamical Semlgroups Using One-dlmensional Quantum
Stochastic Calculus
G.-L. INGOLD, .H GRABERT, Sluggish Decay of Preparation 219
Effects in Low Temperature Quantum systems
VJ
.R JAJTE, Almost Sure Convergence of Iterates of Contractions 231
in Noncommutative L2-Spaces
J.M. LINDSAY, H. MAASSEN, Duality Transform as *-Algebraic 247
Isomorphism
J.M. LINDSAY, K.R. PARTHASARATHY, Rigidity of the Poisson 251
Convolution
H. MAASSEN, A Discrete Entropic Uncertainty Relation 263
B. NACHTERGAELE, Working with Quantum Markov States and their 267
Classical Analogues
H. NARNHOFER, Dynamical Entropy, Quantum K-Systems and 286
Clustering
K.R. PARTHASARATHY, A Continuous Time Version of Stinespring's 296
Theorem on Completely Positive Maps
A. PAS~KIEWICZ, The Topology of the Convergence in Probability 301
in a W -Algebra is Normal
D. PETZ, First Steps Towards a Donsker and Varadhan Theory 311
in Operator Algebras
S. PULMANNOVA, Quantum Conditional Probability Spaces 320
P. ROBINSON, Quantum Diffusions on the Rotation Algebras 326
and the Quantum Hall Effect
J.-L. SAUVAGEOT, Quantum Dirichlet Forms, Differential 334
Calculus and Semigroups
M. SCHORMANN, Gaussian States on Bialgebras 347
G.L. SEWELL, Quantum Macrostatistics and Irreversible 863
Thermodynamics
A.G. SHUHOV, Y.M. SUHOV, Correction to the Hydrodynamical 384
Approximation for Groups of Bogoljubov Transformations
S.J. SUMMERS, Bell's Inequalities and Quantum Field Theory 393
QUANTUM LANGEVIN EQUATION IN mE %7£AK COUPLING LIMIT
Luigi Accardi , Alberto Frigerio , and Lu Yun-Gang
Centro Matematico .V Volterra,
Dipartimento di Matematica, II Universit& di Rome,
00173 La Romanina, Roma, Italy.
Dipartimento di Matematica e Lnformatica,
Universit& di Udine, 33100 Udine, Italy.
On leave from Department of Mathematics,
Beijing Normal University, Beijing, P.R. China.
O. Introduction
This work is part of a series of papers [2,3,4] where, expanding
some heuristic ideas in [11], we develop the theory of the weak
coupling limit for open quantum systems. It has been known for fifteen
years that the reduced dynamics of an open quantum system converges
to a quantum dynamical semigroup in the weak coupling limit (coupling
constant ~ -+ 0 , microscopic time s >-- ~ , with macroscopic time
t = X2s held constant) [18,6]. e)% investigate whether and in which
sense the full time evolution of an open quantum system converges, in
the weak coupling limit, to an evolution driven by quantum Brownian
motion [16,14,15]. In [2] we obtained rigorous results for the time
(k)
evolution operator U ; in [3] we studied the time evolved
t/k 2
(~) CA)+ Ck) CA)
observables j (X) = U (X ~ I)U , and we proved that j
tlA ~ tl~ 2 tl~ = t/A =
converges to a quantum diffusion j [10] governed by a quantum
t
Langevin equation,
Here we wish to present the results of [3] in a self-contained way.
To this end we adopt a method of proof which is simpler than the one
of [3], by reducing the problem to a time-dependent generalization of
the derivation of a quantum dynamical semigroup in the weak coupling
limit [18,6]. However, the price to be paid for this simplification
is that the present method is specific for boson reservoirs with
linear coupling to the system of interest, whereas the method of [3]
is suitable for generalization to the fermion case [4], to more
general interactions, and to the low density limit [9,5].
.1 Notations and Preliminaries
..............................
~e consider a quantum system 5, with associated Hilbert space
and Hamiltonian H, coupled to another quantum system , ~ 5 with
Hilbert space ~" and Hamiltonian 'H , by an interaction kV, where
the coupling constant ~ is assumed to be "small". All Hilbert
spaces in this paper will be understood to be separable. 5 is
spatially confined, meaning that the Hamiltonian H is self-adjoint
in ~ , bounded from below, and such that exp[-~H] is trace class
for all positive ~. ~ 5 is an infinitely extended quasi-free boson
system, meaning that ~ = F(~ ) is the symmetric Fock space over
1
the one-particle Hilbert space ~ , and ~ = H dF(H ) is the
I I
differential second quantization of the one-particle Hamiltonian
H , a non-negative self-adjoint operator in ~ with Lebesgue
1 1
spectrum. e)% shall also assume that there exists a nonzero (nonclosed)
linear subspace K of ~ such that, for all f,g 6 K , we have
1 1 I
I+~ l<f,exp[iH t]g> I dt < + ~ (I.I)
-~ 1
The annihilation and creation operators in ~' corresponding to the
+
test functions f in ~ will be denoted by a(f), a (f); they
1
+
satisfy the CCR [a(f),a (g)] = <f,g> and a(f)~ = O, where
o o
is the Fock vacuum vector.
By doubling the space ~ to ~ ~ ~ , ~ denoting the
1 1 1 1
conjugate space to ~ , this formalism allows us to consider also
I
a representation of the CCR algebra determined by a gauge-invariant
quasi-free state w which is stationary under the time evolution
Q
determined by H'. Specifically, let Q be a positive self-adjoint
operator in ~ , commuting strongly with H and satisfying Q ~ 1,
1 1
and let w be determined by
Q
+ +
W (a(f)a (g)) = <f,~(Q+l)~> , w Ca (g)a(f)) = <f,~(Q-1)~>
Q Q
+
Then a(f) is represented by n Ca(f)) = a(Q f ~ O) + a (0 ~ Q f) ,
Q +
where
Q = [~(Q+I)] , Q = [~(Q-1)] ~ 41.2)
+
+ +
we have indeed <~ ,n (a(f))n (a (g))® > = w (a(f)a (g)) : f,g £ H .
o Q Q ° Q 1
e)% identify ~ and ~ as sets, assuming that ~ is mapped onto
1 1 1 1
by a conjugation commuting with H and with Q , so that
1
Q exp[iH t]f • Q exp[iH t]f = exp[iH t]Q f e exp[-iH t]Q f .
+ 1 1 1 * 1
The interaction V between 5 and S' is assumed to be of the form
n + +
V = i ~ [B ® a (g) - B ® a(g ]) , 41.3)
j=l j j j j
where B ,...,B are bounded operators on H, g ,...,g £ K , and where
1 n 1 n I
[H,B ] = - w B , w > 0 : j = 1,...,n (I.4)
J J J J
<g ,exp[iH t]g > = 0 for all t if j % k (1.5)
j 1 k
Such interactions arise in the so-called rotating wave approximation.
In order to derive a reduced dynamics for the observables of S,
we assume that the initial state of S' is a gauge-invariant quasi-frce
state w Again, by doubling ~ to ~ B( ~ and the set of
Q 1 1 I
indices (I,... ,n) to [I,... ,2n) , we can reduce this to the case
where the initial state of ' S is the rock vacuum: the new expression
of the interaction V becomes
2n + " +
V = i X [B )~ a (g) - B @ a(g )] , (1.6)
j=1 J j j j
~ ÷
where g = Q g S 0 , g = 0 • Q g , and where B = B :
j + j n+j - j n+j j
j = 1,...,n . In this and in the following two Sections, tildes will
be dropped and 2n will be called n again.
Let H = H + H' + XV ( = H @ I + I ~ ' H + AV) be the total
X
Hamiltonian for the composite system 5 + 5' , and let H = H
A=O
o
Let also
(X)
U = exp[iH t] exp[-iH t] : t >- 0 (1.7)
t ° X
be the time evolution operators, in the interaction picture, for
state vectors in H~[ ~ '{}~ Then it has been shown [18,6] that,
for all u , v in IH and for all X in B(~) , one has
CA)+ (k)
lira <u ~ ® ,U (X @ I)U v @ ~ > = <u,T (X)v> , (1.8)
X-~O t/~2 t/X2 t
o o
where (T : t >- O) is a quantum dynamical semigroup (in the sense
t
of [12,17], or a quantum Markovian semigroup in the sense of [1])
whose infinitesimal generator L is given by
n + +
L(X) = ( Z c (w) B X B ) + K X + X K , (1.9)
j=1 j j j j
where
c (w) = - <g ,exp[i(H - w )t]g > dt ( >- 0 ) (1.10)
j j 2 -- j I j j
and where
n
K = - X <g )exp[i(H - w )t] E > dt B B (1.11)
j=l ~ j I j j j j
2. Statement of main result
.............................
In our previous paper [2] we have shown that there is a precise
(X)
sense in which U (defined by (I,7)) converges as A >-- 0 to
tlX ~
the solution U(t) of a quantum stochastic differential equation
(QSDE; in the sense of [16]) of the form
n + +
dU(t) ( ~ [B dA (t) - B dA (t)] + K dr) U(t) (2.1)
j:l j j j j
÷
with U(O) = I ~ where (A (t), A (t) : t ~ 0 , j = 1~...,n] are
J J
mutually independent rock quantum Brownian motions satisfying
+
dA (t) dA (t) = 6 c (w) dt (2.2)
j k jk j j
and where K is given by (1.11) (actually, the proof in [2] is just
for the case n = 1 , but an extension to arbitrary n can be easily
given). In the present paper, like in [3], we shall prove that there
is a precise sense in which
(k) (X)÷ (k)
j (X) :: U (X ® 1) U : X £ B(~4) (2.3)
tlX 2 tlX2 tlA •
converges as ~ -~ 0 to
+
j (X) := U (t) (X ~ 1)U(t) : X E B (CH) , (2.4)
t
where U(t) is the solution of the QSDE (2.1): j is a quantum
t
diffusion (in the sense of [I0]) satisfying the following quantum
Langevin equation:
