Continual Measurements in Quantum Mechanics and ... - CiteSeerX

Continual Measurements in Quantum Mechanics and Quantum Stochastic Calculus

⋆

Alberto Barchielli Politecnico di Milano, Dipartimento di Matematica, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy E-mail: [email protected] Home page: http://www.mate.polimi.it

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.1 1.2 1.3

Three approaches to continual measurements . . . . . . . . . . . . . . . . . . . Quantum stochastic calculus and quantum optics . . . . . . . . . . . . . . . . Some notations: operator spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 3 4

2

Unitary evolution and states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

2.1 Quantum stochastic calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Fock space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Weyl operators and the Bose fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum stochastic integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The unitary system–field evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Hudson–Parthasarathy equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Hamiltonian evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The system–field state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The reduced dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . System observables in the Heisenberg picture . . . . . . . . . . . . . . . . . . . . . . . . The master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Physical basis of the use of QSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The quasi–monochromatic paraxial approximation of the electromagnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximations in the system–field interaction . . . . . . . . . . . . . . . . . . . . . .

5 5 6 9 13 13 15 19 21 21 22 23

3

23 24

Continual measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1 Indirect measurements on SH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Counts of quanta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Measurements of field quadratures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⋆

To appear in Lecture Notes in Mathematics, Springer Version: March 7, 2005

2

Alberto Barchielli

Field observables in the Heisenberg picture . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Characteristic functionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The characteristic functional for the counts of quanta . . . . . . . . . . . . . . . . . The characteristic functional for the field quadratures . . . . . . . . . . . . . . . . Field observables and adapted Weyl operators . . . . . . . . . . . . . . . . . . . . . . . The characteristic functional and the finite dimensional laws . . . . . . . . . . . 3.3 The reduced description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The reduced characteristic operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Instruments and finite–dimensional laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mean values and covariances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Direct detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The detection scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Probabilities for counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Optical heterodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ordinary heterodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Balanced heterodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Physical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

A three–level atom and the shelving effect . . . . . . . . . . . . . . . . 54

4.1 The atom–field dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pure decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stimulating lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The detection process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Bright and dark periods: the V-configuration . . . . . . . . . . . . . . . . . . . . 4.4 Bright and dark periods: the Λ-configuration . . . . . . . . . . . . . . . . . . . . The case of equal detunings: the dark state . . . . . . . . . . . . . . . . . . . . . . . . . . The case of different detunings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

27 29 31 32 32 36 37 37 40 41 43 43 45 46 48 48 50 52 53 55 56 57 58 61 63 63 65

A two–level atom and the spectrum of the fluorescence light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.1 The dynamical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The two-level atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The photon space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The balance equation for the number of photons . . . . . . . . . . . . . . . . . . . . . The phase diffusion model for the laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The master equation and the equilibrium state . . . . . . . . . . . . . . . . . . The parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The equilibrium state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 The detection scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heterodyne detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 The fluorescence spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66 66 67 67 69 69 71 72 73 74 74 75 79

Quantum Continual Measurements

3

The angular distribution of the spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 The total spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

1 Introduction 1.1 Three approaches to continual measurements We speak of continual measurements in the case in which one ore more observables of a quantum system are followed with continuity in time. Traditional presentations of quantum mechanics consider only instantaneous measurements, but continual measurements on quantum systems are a common experimental practice; typical cases are the various forms of photon detection. The statements of a quantum theory about an observable are of probabilistic nature; so, it is natural that a quantum theory of continual measurements give rise to stochastic processes. Moreover, a continually observed system is certainly open. All these things shows that the development and the applications of a quantum theory of continual measurements needs quantum measurement theory, open system theory, quantum optics, operator theory, quantum probability, quantum and classical stochastic processes. . . The first consistent paper treating continual measurements was published in 1969 and concerns counting of quanta [43], but some ideas on quantum counting formulae for photons had already been introduced before [66]. There are essentially three approaches to continual measurements [18, 21, 64]. These approaches have received various degrees of development, any one of them has its own merits and range of applicability, but “morally” all the three approaches are equivalent and one can go from one to the other and this feature is certainly at the bases of the flexibility and interest of the theory. The first approach is the operational one, which is based on positive operator valued measures or (generalized) observables and operation valued measures or instruments [22,23,44]. A variant of this approach is based on the Feynman integral [15, 22, 72, 78]. The second approach is based on quantum stochastic calculus and quantum stochastic differential equations [14, 24] and it is connected to quantum Langevin equations and the notion of input and output fields in quantum optics [54,55]. The last approach is based on (classical) stochastic differential equations and the notion of a posteriori states [21,29] and it is related to some notions appeared in quantum optics: quantum trajectories, Monte–Carlo wave function method, unravelling of master equation [33, 82]. This report is concerned mainly with the second approach, the one based on quantum stochastic calculus. 1.2 Quantum stochastic calculus and quantum optics Quantum stochastic calculus (QSC) [65, 84] was developed originally as a

QSC

4

QSDE

Alberto Barchielli

mathematical theory of quantum noise in open systems and its first applications in mathematical physics were the construction of unitary dilations of quantum dynamical semigroups [65, 84] and of quantum stochastic processes [51]. Soon after it was applied also to measurement theory in quantum mechanics [14, 24]. The “integrators” of QSC are Bose fields (annihilation and creation processes), which play the role of quantum analogues of independent Wiener processes, and some expressions quadratic in the field operators (conservation processes). The starting point for the applications of QSC in quantum optics is to take these Bose fields as an approximation of the electromagnetic field. The explicit introduction of QSC in quantum optics was made in Ref. 54, but the use of the related δ-correlated noise is older [70]. There are various kinds of applications of QSC to quantum optics. The Bose fields are merely considered as a source of noise and quantum stochastic differential equations (QSDE’s) are used for guessing the correct master equation for the system of interest [53, 67, 76, 77]; the fields are used for modelling quantum input and output channels [13,14,17,18,40,54,69]; QSDE’s are used for describing various arrangements for detecting photons [14,16,18,31,32,80]. All these kinds of applications are related; there is not a sharp distinction [18, 55]. A central point in QSC is the ‘quantum stochastic Schrödinger equation’ or Hudson–Parthasarathy equation (2.38), giving the unitary dynamics of a quantum system interacting with the Bose fields. 1.3 Some notations: operator spaces

B(·; ·) B(·)

U(·)

T (·)

S(·) k · k1 partial trace

Let A and B be two Banach spaces; then, we denote by B(A; B) the vector space of linear bounded operators from A into B and we set B(A) := B(A; A). Let K be a complex separable Hilbert space; we denote by U(K) the set of unitary operators on it. Let us recall that in a general quantum theory the states are positive, normalized, linear functionals over a C ∗ - or W ∗ -algebra; we shall consider only the so called normal states, which are represented by trace–class operators. So, we introduce the trace class T (K) := t ∈ B(K) : √ Tr t∗ t < ∞ and the set of statistical operators, or states, S(K) := s ∈ √ T (K) : s ≥ 0, Tr[s] = 1 ; we denote by ktk1 := Tr t∗ t the norm in T (K). Let H and K be two complex separable Hilbert spaces; the partial trace over K is defined by: for t ∈ T (H ⊗ K), TrK {t} ∈ T (H) is the operator satisfying TrH (X TrK {t}) = TrH⊗K {(X ⊗ 1l)t} ,

∀X ∈ B(H) .

(1.1)

Let us end by recalling the definition of core for a selfadjoint operator; see, for instance, [84] p. 64.


5

A closable and densely defined operator T in a Hilbert space H is called symmetric if T ⊂ T ∗ . The operator T is called selfadjoint if T = T ∗ . The operator T is called essentially selfadjoint if its closure is selfadjoint. Let D0 ⊂ D(T ) be a linear manifold and let T0 be the restriction of T to D0 . If T is selfadjoint and the closure of T0 is T , then D0 is called a core for T.

core

2 Unitary evolution and states 2.1 Quantum stochastic calculus We are assuming that the reader is familiar with the main features of QSC, in the version based on the symmetric Fock space, and the Hudson–Parthasarathy equation [65, 84]. In the following we recall a few notions and results of QSC. The Fock space We denote by Γ the symmetric (or boson) Fock space over the “one–particle space” L2 (R+ )⊗Z, where Z is a separable complex Hilbert space ([84] p. 124); we shall see in the physical examples how to choose Z. The space L2 (R+ ) ⊗ Z is naturally identified with L2 (R+ ; Z), so that a vector f in it is a square integrable function from R+ into Z. So, we have Γ := Γsymm L2 (R+ ) ⊗ Z , L2 (R+ ) ⊗ Z ≃ L2 (R+ ; Z). (2.1) Let us denote by e(f ), f ∈ L2 (R+ ; Z), the exponential vectors, whose components in the 0, 1, . . . , n, . . . particle spaces are e(f ) := 1, f, (2!)−1/2 f ⊗ f, . . . , (n!)−1/2 f ⊗n , . . . ; (2.2)

Γ Z

e(f )

the internal product between two exponential vectors is given by he(g)|e(f )i = exphg|f i.

Once normalized, the exponential vectors are called coherent vectors: 1 ψ(f ) := exp − kf k2 e(f ). 2

(2.3) ψ(f ) (2.4)

In particular the vector e(0) ≡ ψ(0) is the Fock vacuum. If M is a dense linear manifold in L2 (R+ ; Z), then the linear span E(M) of the vectors e(f ), f ∈ M, is dense in Γ ; we call exponential domain the set E := E L2 (R+ ; Z) , i.e. the linear span of all the exponential vectors.

E(·) E

6

Γ(s,t) Γ(t

Alberto Barchielli

An important feature of the Fock space Γ is its structure of continuous tensor product. For any choice of the times 0 ≤ s < t let us introduce the spaces Γ(s,t) := Γsymm L2 (s, t) ⊗ Z , Γ(t := Γsymm L2 (t, +∞) ⊗ Z , (2.5)

and for any of such spaces its exponential vectors. Then we have the natural identification ([84] p. 179) Γ ≃ Γ(0,s) ⊗ Γ(s,t) ⊗ Γ(t

f(s,t) , f(t

(here 0 < s < t) based on the factorization of the exponential vectors e(f ) ≃ e f(0,s) ⊗ e f(s,t) ⊗ e f(t ,

(2.6)

(2.7)

where

f(s,t) (τ ) := 1(s,t) (τ ) f (τ ),

f(t (τ ) := 1(t,+∞) (τ ) f (τ ).

Similarly, if P is any orthogonal projection, one has the factorization Γ = Γsymm P L2 (R+ ; Z) ⊗ Γsymm (1l − P )L2 (R+ ; Z) .

(2.8)

(2.9)

The Weyl operators and the Bose fields W(g; U )

The Weyl operator W(g; U ), g ∈ L2 (R+ ; Z), U ∈ U L2 (R+ ; Z) , is the unique unitary operator ([84] Section 20) defined by n o 2 W(g; U ) e(f ) = exp − 21 kgk − hg|U f i e(U f + g) (2.10) or by

W(g; U ) ψ(f ) = exp {i ImhU f |gi} ψ(U f + g) .

(2.11)

W(g; U )−1 = W(g; U )∗ = W(−U ∗ g; U ∗ )

(2.12)

From the definition one obtains easily the inverse

and the composition law W(h; V ) W(g; U ) = exp {−i Imhh|V gi} W(h + V g; V U ) ; CCR

(2.13)

moreover, by particularizing this equation to V = U = 1l, one gets the Weyl form of the canonical commutation relations (CCR): W(h; 1l) W(g; 1l) = W(g; 1l) W(h; 1l) exp {−2i Imhh|gi} .

(2.14)

The Weyl operators allow to introduce some important selfadjoint operators in Γ ([84] Section 20), which play the double role of being the starting point to construct the integrators of QSC and of representing the main observables used in the the theory of continual measurements. Let us collect in a unique theorem Propositions 20.4, 20.7, 20.11, 2.16 and Corollaries 20.5 and 20.6 of Ref. 84.


7

Theorem 2.1. Let us take h ∈ L2 (R+ ; Z); then, the map κ 7→ W(iκh; 1l) is a strongly continuous one parameter group and we denote by Q(h) its Stone generator: W(iκh; 1l) = exp{iκQ(h)}. (2.15)

Q(h)

Moreover, one has (i) Q(h) is essentially selfadjoint in the domain E(M), where M is any dense subset of L2 (R+ ; Z); (ii) E is a core for Q(h); (iii) the linear manifold of all finite particle vectors is a core for Q(h); (iv) E is included in the domain of the product Q(h1 )Q(h2 ) · · · Q(hn ), ∀n, ∀h1 , . . . , hn ∈ L2 (R+ ; Z); (v) [Q(h), Q(g)] e(f ) = {2i Imhh|gi} e(f ), ∀h, g, f ∈ L2 (R+ ; Z).

Let B be a selfadjoint operator in L2 (R+ ; Z) with domain D(B); then, the map κ 7→ W(0; exp{iκB}) is a strongly continuous one parameter group and we denote by λ(B) its Stone generator: W(0; exp{iκB}) = exp{iκλ(B)}.

(2.16)

Moreover, one has (a) E(D(B)) is included in the domain of λ(B); (b) E(D(B 2 )) is a core for λ(B); (c) if B is bounded, λ(B) is essentially selfadjoint in the domain E; (d) i[λ(B1 ), λ(B2 )] e(f ) = λ (i[B1 , B2 ]) e(f ), for any two bounded selfadjoint operators B1 , B2 and ∀f ∈ L2 (R+ ; Z); (e) E is included in the domain of the product T1 T2 · · · Tn , where Ti = Q(hi ), hi ∈ L2 (R+ ; Z), or Ti = λ(Bi ), Bi = Bi∗ ∈ B L2 (R+ ; Z) .

For any h ∈ L2 (R+ ; Z) and any selfadjoint operator B in L2 (R+ ; Z) let us set λ(B, h) := W(−h; 1l) λ(B) W(h; 1l) . (2.17)

Then, the operator λ(B, h) is the generator of the unitary group κ 7→ W(−h; 1l)W(0; exp{iκB})W(h; 1l)

≡ exp i Im eiκB h h W

eiκB − 1l h; eiκB

(2.18)

and it is essentially selfadjoint on the linear manifold generated by e(f −h) : f ∈ D(B 2 ) . When B is also bounded, E is a core for λ(B, h) and, on the exponential domain E, one has λ(B, h) = λ(B) + a(Bh) + a† (Bh) + hh|Bhi1l . The operators a(·) and a† (·) are defined here below in eq. (2.20).

(2.19)

λ(B)

8

a(h), a† (h)

Alberto Barchielli

By defining a(h) =

1 Q(h) + iQ(ih) , 2

a† (h) =

1 Q(h) − iQ(ih) , 2

(2.20)

one obtains two mutually adjoint operators, satisfying ([84] Proposition 20.12) the eigenvalue relation a(h) e(f ) = hh|f i e(f ) (2.21) and, at least in the domain E, the CCR [a(h), a(g)] = [a† (h), a† (g)] = 0 ,

[a(h), a† (g)] = hh|gi .

(2.22)

So, we recognize the annihilation and creation operators and we call them, collectively, the (smeared) Bose fields; in quantum optics the two selfadjoint operators Q(h) and Q(ih) are sometimes referred to as two conjugated field quadratures. If B ∈ B L2 (R+ ; Z) is bounded but not selfadjoint, one defines λ(B) := 1 λ 12 (B + B ∗ ) +iλ 2i (B − B ∗ ) . All matrix elements on exponential vectors and commutation relations involving Q(h), a(h), a† (h), λ(B) are deduced from the properties of the Weyl operators and of the exponential vectors and are given in Ref. 84 Section 20. Moreover, also the linear manifold of all finite particle vectors is contained in the domains of Q(h), a(h), a† (h), λ(B) ([84] Proposition 20.14). The annihilation, creation and conservation processes c.o.n.s. {zk }, fk (t) Ak (t), A†k (t)

Let us fix a complete orthonormal system (c.o.n.s.) {zk , k ≥ 1} in Z and for any f ∈ L2 (R+ ; Z) let us set fk (t) := hzk |f (t)i. We denote by Ak (t), A†k (t), Λkl (t) the annihilation, creation and conservation processes associated with such a c.o.n.s. ([65] Sect. 2): A†k (t) := a† zk ⊗ 1(0,t) , Λkl (t) := λ (|zk ihzl |) ⊗ 1(0,t) ;

Ak (t) := a zk ⊗ 1(0,t) , Λkl (t)

(2.23a) (2.23b)

for these processes one has

Ak (t) e(f ) =

Z

t

fk (s) ds e(f ) ,

(2.24a)

gk (s) ds he(g)|e(f )i,

(2.24b)

gk (s) fl (s) ds he(g)|e(f )i.

(2.24c)

0

he(g)|A†k (t)e(f )i = he(g)|Λkl (t)e(f )i =

Z

t

0

Z

0

t

Let us recall that by construction these operators are defined at least on E and on this domain A†k (t) is the adjoint of Ak (t); moreover, Ak (t) + A†k (t),


9

i A†k (t) − Ak (t) , Λkk (t) are essentially selfadjoint on E. Another form of the CCR follows from (2.23a) and (2.22): on the exponential domain, and on the finite particle vectors, one has [Ak (t), Al (s)] = [A†k (t), A†l (s)] = 0,

[Ak (t), A†l (s)] = δkl min{t, s}. (2.25)

In theoretical physics it is usual to write formally Z t Z t Z t Ak (t) = ak (s) ds , A†k (t) = a†k (s) ds , Λkl (t) = a†k (s)al (s) ds , 0

0

0

(2.26)

where the “Bose fields” ak (t), a†k (t) satisfies the (heuristic) CCR [ak (t), a†l (s)] = δkl δ(t − s),

[ak (t), al (s)] = [a†k (t), a†l (s)] = 0 .

(2.27)

Quantum stochastic integrals The system space We are interested in a quantum system interacting with the Bose fields we have introduced; this quantum system is described in a complex separable Hilbert space H, called the system space or the initial space ([84] p. 179); let us call this quantum system “system SH ” or simply “the system”. Operators acting in H are extended to H ⊗ Γ by the convention that they act as the identity on Γ ; the tensor product with the identity is not always indicated. A similar extension is understood for operators acting in Γ . So, K ∈ B(H) or K ⊗ 1l, Ak (t) or 1l ⊗ Ak (t) are the same.

H SH

Adapted processes An adapted process is a time dependent family of operators {L(t), t ≥ 0}, such that L(t) acts trivially as the identity on Γ(t and possibly non trivially in H ⊗ Γ(0,t) ; an adapted process is something containing the fields only up to time t. In the case of a bounded adapted process, this simply means L(t) ∈ B H ⊗ Γ(0,t) ; in the general case the definition is the following.

Definition 2.1 ([84] p. 180). Let D and M be dense linear manifolds in H and L2 (R+ ; Z), respectively, such that 1(s,t) f ∈ M whenever f ∈ M for all 0 ≤ s < t < ∞. Denote by D ⊗ E(M) the linear manifold generated by all vectors of the form u ⊗ e(f ), u ∈ D, f ∈ M. A family {L(t), t ≥ 0} of operators in H ⊗ Γ is an adapted process with respect to (D, M) if

⊗ adapted process

(i) for any t, the domain of L(t) contains D ⊗ E(M); (ii) L(t)u⊗e(f(0,t) ) ∈ H⊗Γ(0,t) and L(t)u⊗e(f ) = L(t)u ⊗ e(f(0,t) ) ⊗e(f(t ) for all t ≥ 0, u ∈ D, f ∈ M. It is said to be regular if, in addition, the map t 7→ L(t)u ⊗ e(f ) from R+ into H ⊗ Γ is continuous for every u ∈ D, f ∈ M.

regular

10

Alberto Barchielli

It is convenient to fix M once for all; we follow the choice of Ref. 71:

M dim f

M := f ∈ L2 (R+ ; Z) ∩ L∞ (R+ ; Z) : fk (t) ≡ 0

for all but a finite number of indices k . (2.28) For f ∈ M let us set dim f := max k fk is a non-zero vector in L2 (R+ ) . Let us note that the definitions of M and dim f depend on the initial choice of the c.o.n.s. {zk } in Z and that, being M dense in L2 (R+ ; Z), then E(M) is total in Γ . Quantum stochastic integrals and Ito table In QSC integrals of “Ito type” with respect to dAk (t), dA†k (t), dΛkl (t) are defined ([84] pp. 188–190, 224–225); the integrands are adapted processes with some conditions to assure the existence of the quantum stochastic integrals. We shall use the class of integrands given in the definition below. It is the one used in Ref. 71 and it allows to give meaning to all the integrals we need; it is a bit larger of the one introduced by Parthasarathy [84], but all the results of Ref. 84 continue to hold.

stochastically integrable

Definition 2.2 ([65] Proposition 3.2; [84] pp. 189, 221–222, 224; [71]). A family {Lkl , k, l ≥ 0} of (H, M) adapted processes is said to be stochastically integrable if, for all t ≥ 0, l ≥ 0, u ∈ H, f ∈ M, Z tX ∞

0 k=0

L(M)

kLkl (s)u ⊗ e(f )k2 1 + kf (s)k2 ds < +∞ .

We denote by L(M) the class of stochastically integrable families of (H, M) adapted processes. Note that M and, so, the class of stochastically integrable processes L(M) depend on the initial choice of the c.o.n.s. {zk } in Z. The definition of quantum stochastic integral goes trough a suitable limit on the integral of sequences of “simple processes” ([84] Section 27); then, the following result holds. Proposition 2.1 ([84] Proposition 27.1). Let {Lkl , k, l ≥ 0} ∈ L(M), X(0) ∈ B(H); then X(t) := X(0) +

Z t ∞ X L00 (s)ds + Lk0 (s)dA†k (s) 0

k=1

+

∞ X l=1

L0l (s)dAl (s)

+

∞ X

k,l=1

Lkl (s)dΛkl (s)

is a regular (H, M) adapted process and, ∀u ∈ H, ∀f ∈ M, ∀t ≥ 0,

(2.29)

Quantum Continual Measurements 2

k[X(t) − X(0)] u ⊗ e(f )k ≤ 2 exp ×

dim Xf l=0

Z tX ∞

0 k=0

Z

11

1 + kf (s)k2 ds

t

0

kLkl (s)u ⊗ e(f )k2 1 + kf (s)k2 ds .

(2.30)

The main practical rules to manipulate the quantum stochastic integrals and their products are the facts that 1. dAk (t), dA†k (t), dΛkl (t) commute with adapted processes at time t, so that they can be shifted towards the right or the left, according to the convenience; 2. the products of the fundamental differentials satisfy the Ito table dAk (t) dA†l (t) = δkl dt , dΛkr (t) dA†l (t)

=

dAk (t) dΛrl (t) = δkr dAl (t) ,

δrl dA†k (t) ,

dΛkr (t) dΛsl (t) = δrs dΛkl (t) ; (2.31)

all the other products and the products involving dt vanish; 3. dAk (t) e(f ) = dt fk (t) e(f ), he(f )| dA†k (t) = fk (t) dt he(f )|, † dΛkl (t) e(f ) = dAk (t) fl (t) e(f ). By means of these rules the matrix elements on exponential vectors of quantum stochastic integrals can be computed and products of quantum stochastic integrals can be differentiated (quantum Ito’s formula); see [84] pp. 221–224. To be more precise, the following results hold. Proposition 2.2 ([84] Corollary 27.2). Let {Lkl , k, l ≥ 0} , {Mlk , k, l ≥ 0} ∈ L(M), X(0) , Y (0) ∈ B(H); let X(t) be defined by eq. (2.29) and Y (t) be defined in a similar way in terms of the processes Mlk . Then, for all u, v ∈ H, f, g ∈ M the matrix elements hv ⊗ e(g)|X(t)u ⊗ e(f )i, hv ⊗ e(g)|Y (t)u ⊗ e(f )i, hY (t)v ⊗ e(g)|X(t)u ⊗ e(f )i are just the ones that one can compute by means of the practical rules given above, i.e.

v ⊗ e(g) [X(t) − X(0)] u ⊗ e(f ) = × Lk0 (s) +

∞ X l=1

L0l (s)fl (s) +

Z

∞ D X ds v ⊗ e(g) L00 (s) + gk (s)

t

0

∞ X

k,l=1

k=1

E gk (s)Lkl (s)fl (s) u ⊗ e(f ) ,

(2.32)

hY (t) v ⊗ e(g)|X(t) u ⊗ e(f )i − hY (0) v ⊗ e(g)|X(0) u ⊗ e(f )i Z t = ds hY (s) v ⊗ e(g)|L00 (s) u ⊗ e(f )i 0

+ hM00 (s) v ⊗ e(g)|X(s) u ⊗ e(f )i +

∞ X

hM0k (s) v ⊗ e(g)|Lk0 (s) u ⊗ e(f )i

k=1

12

Alberto Barchielli

+

∞ X

k=1

+

hMk0 (s) v

+

gk (s) hY (s) v ⊗ e(g)|Lk0 (s) u ⊗ e(f )i

⊗ e(g)|X(s) u ⊗ e(f )i +

∞ X l=1

hMkl (s) v

⊗ e(f )i

∞ X hY (s) v ⊗ e(g)|L0l (s) u ⊗ e(f )i + hM0l (s) v ⊗ e(g)|X(s) u ⊗ e(f )i l=1

+ +

⊗

e(g)|Ll0 (s) u

∞ X

k,l=1

∞ X

hM0k (s) v k=1

⊗

e(g)|Lkl (s) u

⊗ e(f )i fl (s)

gk (s) hY (s) v ⊗ e(g)|Lkl (s) u ⊗ e(f )i + hMkl (s) v ⊗ e(g)|X(s) u ⊗ e(f )i +

∞ X r=1

hMkr (s) v

⊗

e(g)|Lrl (s) u

⊗ e(f )i fl (s) .

(2.33)

Moreover, the quantum stochastic integral (2.29) is uniquely determined by the matrix elements (2.32). To handle a product Y (t)X(t) is a more delicate problem; in general we do not even know if H ⊗ E(M) is included in the domain of this product.

adjoint pair

Definition 2.3. Let {Lkl , k, l ≥ 0}, {L†kl , k, l ≥ 0} ∈ L(M) be such that hv ⊗ e(g)|Llk (t)u ⊗ e(f )i = hL†kl (t)v ⊗ e(g)|u ⊗ e(f )i, ∀k, l, t, ∀u, v ∈ H, ∀f, g ∈ M; then, {Lkl , k, l ≥ 0}, {L†kl , k, l ≥ 0} is called an adjoint pair in L(M). Proposition 2.3 ([84] Proposition 25.12, [71] p. 518). Let {Lkl , k, l ≥ 0}, {L†kl , k, l ≥ 0} be an adjoint pair in L(M), X(0) ∈ B(H), X † (0) = X(0)∗ ; let X(t) be defined by eq. (2.29) and X † (t) be defined in a similar way in terms of L†kl , X † (0). Then, X(t)∗ = X † (t), ∀t ≥ 0, on H ⊗ E(M). Proposition 2.4 ([84] Proposition 25.26). Let {Lkl , k, l ≥ 0} , {Mlk , k, l †k k ≥ 0} , {M †k l , k, l ≥ 0} ∈ L(M), with {Ml , k, l ≥ 0} , {M l , k, l ≥ 0} an adjoint pair, X(0) , Y (0) ∈ B(H); let X(t) be defined by eq. (2.29) and Y (t) be defined in a similar way in terms of Mlk , Y (0). Let us define, ∀k, l ≥ 0, ∀t ≥ 0, ∞ X Flk (t) := Mlk (t)X(t) + Y (t)Lkl (t) + Mjk (t)Ljl (t) . (2.34) j=1

Suppose that i) Y (t)X(t), t ≥ 0, in an (H, M ) adapted process, ii) {Flk , k, l ≥ 0} ∈ L(M).


13

Then, on H ⊗ E(M), one has Y (t)X(t) = Y (0)X(0) +

Z t ∞ X F0k (s)dA†k (s) F00 (s)ds + 0

+

k=1

∞ X

Fl0 (s)dAl (s)

+

l=1

∞ X

Flk (s)dΛkl (s)

k,l=1

.

(2.35)

Parthasarathy gives the proof of this proposition only in the case of finitely many integrators, but nothing changes in the case of infinitely many integrators and of integrands in L(M); the existence of the adjoint pair is needed in the proof, but it does not appear in the final statement. The meaning of Proposition 2.4 is that, under the hypotheses given, Y (t)X(t) can be differentiated according the practical rules of QSC. 2.2 The unitary system–field evolution The Hudson–Parthasarathy equation The ingredients of the Hudson–Parthasarathy equation are system operators and fields. We consider the simplest version of this equation: only bounded system operators are involved. This is not enough for all physical applications, but includes significant cases and allows to develop a general theory which gives an idea of what could be done even in other cases. The system operators Let H, Rk , k P ≥ 1 and Skl ,Pk, l ≥ 1, be bounded operators in H such that ∗ ∗ H ∗ = H and j Sjk Sjl = j Skj Slj = δkl ; if Z isP infinite dimensional, the set of indices is infinite and the previous series and k Rk∗ Rk are assumed to be strongly convergent P to bounded operators. From these conditions, we have that also the series k Rk∗ Skl is strongly convergent to a bounded operator. It is useful to construct from H, Rk , Skl three operators S ∈ U(H ⊗ Z), by Ru =

R ∈ B(H; H ⊗ Z), X k

or, equivalently,

K ∈ B(H)

H Rk Skl

(2.36a) R, S, K

(Rk u) ⊗ zk ,

hv ⊗ zk |Rui = hv|Rk ui ,

∀u ∈ H ,

(2.36b)

∀ u, v ∈ H ,

(2.36c)

14

Alberto Barchielli

S :=

X kl

K := −iH − C(f )

Skl ⊗ |zk ihzl | ,

(2.36d)

1X ∗ 1 Rk Rk ≡ −iH − R∗ R . 2 2

(2.36e)

k

Another useful operator, for f ∈ Z, is C(f ) : H → H ⊗ Z ,

C(f )u = u ⊗ f ,

∀u ∈ H .

(2.36f)

By the positions (2.36) we have also the useful relations X ∗ gk (s)Rk = C g(s) R , X kl

∗ gk (s) (Skl − δkl ) fl (s) = C g(s) (S − 1l)C f (s) , X kl

The evolution equation U (t)

(2.37a)

k

(2.37b)

Rk∗ Skl fl (s) = R∗ SC f (s) .

(2.37c)

Theorem 2.2 ([84] Proposition 27.5 p. 225, Theorem 27.8 p. 228). In the hypotheses above, there exists a unique (H, M) adapted process U (t) satisfying the initial condition U (0) = 1l and the QSDE X X dU (t) = Rk dA†k (t) + (Skl − δkl ) dΛkl (t) k

kl

−

X kl

Rk∗ Skl dAl (t) + K dt U (t).

(2.38)

Moreover, U (t) is uniquely extended to a unitary process, which turns out to be strongly continuous in t. The adjoint process U (t)∗ is strongly continuous and it is the unique unitary adapted process satisfying U (0)∗ = 1l and the adjoint equation X X ∗ (Skl − δkl ) dΛlk (t) Rk∗ dAk (t) + dU (t)∗ = U (t)∗ kl

k

−

X

∗ Skl Rk

kl

dA†l (t)

+ K dt . ∗

(2.39)

By the practical rules of QSC (Proposition 2.2) we get from (2.38): ∀u, v ∈ H, ∀f, g ∈ M Z t D nX hv ⊗ e(g)|U (t) u ⊗ e(f )i − hv|ui exp{hg|f i} = ds v ⊗ e(g) gk (s)Rk 0

+

X kl

gk (s) (Skl − δkl ) fl (s) −

X kl

Rk∗ Skl fl (s)

k

o E + K U (s) u ⊗ e(f ) ;

(2.40)


15

it is also true that this equation uniquely determines U . Corollary 2.1 ([56] Corollary 3.2 p. 26). The solution U of (2.38) is the unique unitary adapted process satisfying (2.40) for every t ≥ 0, u, v ∈ H, f, g ∈ M. Proof. Let U ′ be a unitary adapted process satisfying (2.40) and let us define e ′ (t) := 1l + U

Z t X 0

Rk dA†k (s) +

k

X kl

(Skl − δkl ) dΛkl (s) −

X kl

Rk∗ Skl dAl (s) + K ds U ′ (s).

We see that, by the hypotheses on the system operators and the unitarity of U ′ (s), the coefficients Rk U ′ (s), . . . satisfy the conditions in Definition 2.2; e ′ (t) is well defined by Proposition 2.1 and its matrix elements on then, U exponential vectors are given by Proposition 2.2. So, for every t ≥ 0, u, v ∈ H, e ′ (t) u ⊗ e(f )i = hv ⊗ e(g)|U ′ (t) u ⊗ e(f )i; having f, g ∈ M, we have hv ⊗ e(g)|U e ′ (t) and U ′ (t) are equal equal matrix elements on a total set of vectors, U e ′ (t) satisfies the QSDE (2.38). The uniqueness of the and this implies that U e ′ (t) = U ′ (t) = U (t). solution of (2.38) implies U ⊓ ⊔ The Hamiltonian evolution

As soon as Hudson and Parthasarathy introduced their equation, Frigerio and Maassen independently realized that each unitary solution U is naturally associated to a strongly continuous one–parameter unitary group V [50, 51, 73,74]. This is important for physical applications: a strongly continuous one– parameter unitary group is what we want in quantum mechanics to give the dynamics of an isolated system, here system SH and fields. To obtain this result we need to enlarge the Fock space Γ ; it is convenient to consider this ampliation of Γ only in this subsection, because it has no effect in all the rest of the paper. With the notations of eq. (2.5), we have Γ ≡ Γ(0 ; now, let us introduce the spaces Γt) := Γsymm L2 (−∞, t) ⊗ Z and Γe := Γsymm L2 (R) ⊗ Z ≡ Γ0) ⊗ Γ(0 . With the usual convention of not to write the tensor products with the identity, the solution U (t) of eq. (2.38) can be understood as a unitary operator on H ⊗ Γe. Then, we introduce the strongly continuous one–parameter unitary group θ of the shift operators on L2 (R; Z) and its second quantization Θ on Γe: for every t ∈ R θt f (r) = f (r + t), ∀f ∈ L2 (R; Z), (2.41a) Θt e(f ) = e(θt f ),

∀f ∈ L2 (R; Z);

(2.41b)

Γt) , Γe

θt , Θt

16

Alberto Barchielli

we extend Θt to the space H ⊗ Γe. Now one can prove the cocycle property (2.42) for U , which Accardi already showed to give rise to groups [1, 7].

Theorem 2.3 ([50, 51, 73, 74]). Let Θ be the strongly continuous one– parameter unitary group defined by (2.41) and let U be the solution of the QSDE (2.38) with system operators satisfying the conditions of Theorem 2.2. Then, they are related by the cocycle property (of U with respect to Θ) U (s + t) = Θs∗ U (t) Θs U (s), Vt

U (t, s)

∀s, t ≥ 0,

and the family of unitary operators V = {Vt }t∈R , defined by ( Θt U (t), if t ≥ 0, Vt = ∗ U (|t|) Θt , if t ≤ 0,

(2.42)

(2.43)

is a strongly continuous one–parameter unitary group. Moreover, the two–parameter family of unitary operators U (t, s) := Θt∗ Vt−s Θs ≡ Θs∗ U (t − s)Θs ,

s ≤ t,

(2.44)

is strongly continuous in t and s and satisfies the composition law r ≤ s ≤ t.

U (t, s) U (s, r) = U (t, r) ,

(2.45)

The operator U (t, s) is adapted to H ⊗ Γ(s,t) , i.e. it acts as the identity on Γs) ⊗ Γ(t and with respect to t it satisfies the Hudson–Parthasarathy equation (2.38) with initial condition U (s, s) = 1l. Proof. Let us note that eq. (2.38) gives also Z t+s X X Rk dA†k (r) + U (t + s) − U (s) = (Skl − δkl ) dΛkl (r) s

k

kl

−

X

Rk∗ Skl

kl

dAl (r) + K dr U (r) .

(2.46)

Let us consider Xs (t) := Θs U (s + t)U (s)∗ Θ−s ; note that Xs (0) = 1l. By Proposition 2.2, the definition of Θ (2.41), the previous equation and a change of integration variable we get D

v ⊗ e(g) Xs (t) − 1l u ⊗ e(f ) = v ⊗ e(θ−s g) Z t+s nX X dr gk (r − s) Rk + gk (r − s) (Skl − δkl ) fl (r − s) s

−

X kl

k

kl

o E Rk∗ Skl fl (r − s) + K U (r)U (s)∗ u ⊗ e(θ−s f )

Z t nX X dτ = v ⊗ e(g) gk (τ ) Rk + gk (τ ) (Skl − δkl ) fl (τ ) D

0

k

−

X kl

kl

Rk∗ Skl fl (τ )

o E + K Xs (τ ) u ⊗ e(f ) .


17

Therefore Xs (t) satisfies eq. (2.40) as U (t) and, by Corollary 2.1, U (t) = Xs (t), which is equivalent to the cocycle property (2.42). Let us prove now the group property for V ; from the definition (2.43) one has the unitarity of Vt and Vt∗ = V−t ,

∀t ∈ R .

(2.47)

From the cocycle property (2.42) and the fact that Θ is a group one gets, ∀t, s ≥ 0, Vt Vs = Θt U (t)Θs U (s) = Θt+s Θs∗ U (t)Θs U (s) = Θt+s U (t + s) = Vt+s . (2.48) All the other combinations of positive and negative times can be examinated and give the same result. For instance, for s ≤ 0, t + s ≥ 0, one has from (2.48) Vt+s V−s = Vt , which is equivalent to the group property Vt+s = Vt Vs due to (2.47). Being V a unitary group, it is enough to prove its strong continuity in 0, which follows from the unitarity and the strong continuity of U and Θ. For t ≥ 0 and Υ ∈ H ⊗ Γe we have

Vt − 1l Υ = Θt U (t) − 1l Υ ≤ Θt U (t) − 1l Υ + Θt − 1l Υ

as t ↓ 0 . = U (t) − 1l Υ + Θt − 1l Υ → 0

For t ≤ 0, we have

Vt − 1l Υ = U (|t|)∗ Θt − 1l Υ

≤ U (|t|)∗ Θt − 1l Υ + U (|t|)∗ − 1l Υ

= Θt − 1l Υ + U (|t|) − 1l Υ → 0

as t ↑ 0 .

From U (t, s) = Θs∗ U (t − s)Θs and Corollary 2.1 one has immediately that U (t, s) is adapted to H ⊗ Γ(s,t) and satisfies the QSDE (2.38). From U (t, s) = Θt∗ Vt−s Θs and the fact that Θ and Vt are strongly continuous unitary groups one has immediately the composition law (2.45) and, by using also U (t + r, s) − U (t, s) = Θt∗ (Θr∗ Vr − 1l)Vt−s Θs and U (t, s + r) − U (t, s) = ⊓ ⊔ Θt∗ Vt−s (Vr∗ Θr − 1l)Θs , the strong continuity in t and s. The interaction picture Theorem 2.3 is essential for the interpretation of the whole construction. The group V (2.43) is the reversible evolution of the isolated system SH plus Bose fields. The free evolution of the fields is supposed to be given by Θ, and hence, physically, the degree of freedom described by L2 (R) is thought to be the conjugate moment of the one–particle energy. Then, U (t) = U (t, 0) = Θt∗ Vt is the evolution operator giving the state dynamics from time 0 to time t of the whole system in the interaction picture with respect to the free field dynamics.

18

Alberto Barchielli

In quantum mechanics probabilities, mean values, . . . can be reduced to expressions like as hB, ˜sit := TrH⊗Γe {BVt˜sVt∗ } , (2.49)

where ˜s ∈ S(H ⊗ Γe) is the initial state at time zero and B ∈ B(H ⊗ Γe). To pass to the interaction picture means to write (2.49) as hB, ˜sit = TrH⊗Γe {B(t)U (t)˜sU (t)∗ } ,

B(t) := Θt∗ BΘt .

(2.50)

If B(t) ∈ B(H ⊗ Γ ) at a certain time t (i.e. it acts as the identity on Γ0) ) and ˜s = s0) ⊗ s with s0) ∈ S(Γ0) ) and s ∈ S(H ⊗ Γ ), we can get ride of Γ0) and we write hB, ˜sit = TrH⊗Γ {B(t)U (t)sU (t)∗ } . (2.51)

This is the case when B ∈ B(H) (one gets B(t) = B, ∀t); but this is not the only possible case and also field observables can be involved. By (2.24a) and (2.41) one obtains Θs∗ Ak (t)Θs = Ak (t + s) − Ak (s) ,

(2.52)

which shows that the fields Ak (t), and therefore also A†k (t), Λkl (t), are already in the interaction picture. This conclusion is clearer if we use the heuristic notation (2.26) and the “energy representation” of the fields: Z +∞ 1 eiωt ak (t)dt . (2.53) a ˆk (ω) = √ 2π −∞ Indeed we get

Θs∗ Ak (t)Θs

=

Z

0

t

1 dτ √ 2π

Z

+∞

dω e−iω(τ +s) a ˆk (ω) ,

(2.54)

−∞

Θs∗ a ˆk (ω)Θs = e−iωs a ˆk (ω) .

(2.55) e From the next subsection on, we shall go back to use Γ and not Γ , because automatically we work in the interaction picture and consider only states and observables adapted to (0, +∞) in the sense above. The Hamiltonian operator Being a strongly continuous unitary group, V is differentiable on the domain of its generator, it gives rise to a Schrödinger equation of usual type and a selfadjoint operator with the role of Hamiltonian exists. The characterization of this Hamiltonian required some effort; the results are in Refs. 34–38,57, 58. Also before these works, it was clear that this Hamiltonian must be unbounded either from above, either from below; in this sense, the dynamics V , or, equivalently, U , must necessarily be an approximation to a “true” dynamics with a physical Hamiltonian bounded from below. Let us also stress that the unitary groups Θ and V are strongly differentiable on the dense domains of their Hamiltonians; being the two Hamiltonians unbounded, there is no surprise that the product U of the two groups be not differentiable. From this point of view the surprise is that U satisfies a closed equation, the QSDE (2.38).


19

2.3 The system–field state Once the dynamical equation (2.38) is constructed, one needs an initial state

s

s ∈ S(H ⊗ Γ ) . From a mathematical point of view any statistical operator is a possible initial state; from a physical point of view the whole theory is an approximation and whether this approximation is good or not can depend also on the initial state. So, the interpretation of the theory depends not only on the dynamics (2.36)– (2.38), but also on the initial state; moreover, such a state must be sufficiently simple to allow computations. The usual choice for the initial state is a factorized state s = ρ0 ⊗ σ ,

ρ0 ∈ S(H) ,

σ ∈ S(Γ ) .

(2.56)

Let us use the notation η(f ) := |ψ(f )ihψ(f )| ,

η(f ) f ∈ L2 (R+ ; Z) ,

(2.57)

for a pure coherent state. The coherent states are the only statistical operators in S(Γ ) which enjoy the factorization property σ = σ(0,s) ⊗ σ(s , σ(0,s) ∈ S(Γ(0,s) ), σ(s ∈ S(Γ(s ), ∀s ≥ 0; indeed, from (2.7) one has η(f ) = η f(0,s) ⊗ η f(s , ∀s ≥ 0 , (2.58) with η f(0,s) ∈ S(Γ(0,s) ) , η f(s ∈ S(Γ(s ) . In the following, we list and comment possible choices for σ.

S1.

σ = η(0),

the vacuum [65].

This is the case when the fields act essentially as a reservoir. The reduced dynamics of system SH turns out to be time homogeneous and to be given by a quantum dynamical semigroup. S2.

σ = η(f ),

a coherent state for the field [18].

Now the fields provide also a coherent source which stimulates the system SH . This is the typical choice when a “perfectly” coherent laser is present; a coherent and monochromatic laser could be represented by η(f ) with f (t) = 1[0,T ] (t) e−iωt λ ,

λ∈Z,

ω > 0,

(2.59)

where T is a very large time; eventually, one takes T → +∞ at the end in the physical quantities. Equation (2.58) implies a related factorization property for the state at time t in the interaction picture, because U is an adapted process: U (t) [ρ0 ⊗ η(f )] U (t)∗ = U (t) ρ0 ⊗ η f(0,t) U (t)∗ ⊗ η f(t , (2.60) ∗ with U (t) ρ0 ⊗ η f(0,t) U (t) ∈ S(Γ(0,t) ) , η f(t ∈ S(Γ(t ) .

20

S3. Ec

Alberto Barchielli

σ = Ec η(f )

a mixture of coherent states [28].

Here we mean that there is a random variable f in a probability space (Ω c , F c , P c ) with values in L2 (R+ ; Z). Then, the expectation is given by Z c E η(f ) = η f (·; w) P c (dw), (2.61) Ωc

where the integral can be understood in the topology induced in T (Γ ) by the duality with B(Γ ); being η(f ) a pure state, this means: ∀A ∈ B(Γ ), Z

TrΓ {Aσ} = (2.62) ψ f (·; w) A ψ f (·; w) P c (dw). Ωc

It is always possible to think f as a stochastic process defined for times in R+ with values in Z and with trajectories in L2 (R+ ; Z); moreover, by taking a filtration containing its natural one, it is always possible to take as f an adapted, or non anticipating, process. So we complete S3 with S3’. (Ω c , F c , P c ) is a probability space with a filtration {Ftc , t ≥ 0}, i.e. Ftc is a σ-algebra with Fsc ⊂ Ftc ⊂ F c for all times 0 ≤ s ≤ t; f (t), t ≥ 0, is a progressively measurable process, and hence adapted, with f (t; w) ∈ Z, f (·; w) ∈ L2 (R+ ; Z), for w ∈ Ω c . Now the factorization property (2.60) does not hold, but we have only U (t) [ρ0 ⊗ σ] U (t)∗ = Ec U (t) ρ0 ⊗ η f(0,t) U (t)∗ ⊗ Ec η f(t Ftc (2.63) As an example take the “phase diffusion model” of a laser, used in Ref. 28 in the study of the fluorescence spectrum of a two–level atom: the field state is given by S3, where f is the process f (t) = e−i(ωt+

√ B W (t))

1(0,T ) (t) λ ,

λ∈Z,

ω > 0,

B ≥ 0;

(2.64)

W (t) is a real standard Wiener process canonically realized in the Wiener probability space (Ω c , F c , P c ). In the case S3 we have a demixture {η f (·; w) , P c (dw)} of a statistical operator σ into pure states η(f ). Often in quantum mechanics the point of view is taken that in a single, individual experiment a pure state is realized; then, if there is not a perfect control of the preparation, in replicas of the experiment other pure states are realized according to some probability law: mixed states arise due to our imperfect knowledge of the initial state. This interpretation is not always justified, certainly not in the typical situations of statistical mechanics when thermal states and thermodynamical limits enter into play. Another difficulty is that, given a mixed state σ, there are infinitely many demixtures, one of them being the one determined by its eigenvalues and eigenvectors. However, in our case the demixture {η f (·; w) , P c (dw)} can be thought as a special one among the possible demixtures of σ, as the physical


21

one. For instance in the example (2.64), we have a laser nearly monochromatic, nearly coherent, but with a fluctuating phase, fluctuating in time and from an experiment to another; t 7→ f (t; w) gives the history of the laser in a single experiment, while from an experiment to another it is w to change. In quantum optics the mixtures of coherent states S3, and the particular cases S2 and S1, are called classical states; all the other states are called quantum states. Among the quantum states are the Scr¨ odinger cats, which are quantum superpositions of coherent vectors, such as |αψ(f )+βψ(g)ihαψ(f )+ βψ(g)|. To my knowledge, quantum states have never been used as initial states in the “quantum stochastic framework”, when the QSC is based on Fock space; some “non classical” situations have been approached by using versions of QSC based on non Fock representations of the CCR (thermal and squeezed noise) [54, 55]. 2.4 The reduced dynamics System observables in the Heisenberg picture Let us consider any observable X ∈ B(H) of the quantum system SH ; X can be in particular a selfadjoint operator or even a projection operator. . . We already discussed the interaction picture, but we can use also the Heisenberg picture; being Vt the unitary group giving the system field dynamics, the observable X in the Heisenberg picture becomes X(t) = Vt∗ (X ⊗ 1l)Vt . Recalling that Vt = Θt U (t) and that Θt commutes with X ⊗1l, we obtain X(t) = U (t)∗ (X ⊗1l)U (t). By the rules of QSC, X(t) can be differentiated; the result is a “quantum stochastic” Heisenberg equation. Proposition 2.5 ([84] Corollary 27.9). Let us define jt (X) := U (t)∗ (X ⊗ 1l)U (t) ,

∀X ∈ B(H) ;

jt (2.65)

then, we have ∞ X djt (X) = jt L′0 [X] dt + jt Rk [X] dA†k (t) k=1

+

∞ X

k=1

where

∞ X jt Skl [X] dΛkl (t) , jt Rk [X ] dAk (t) + ∗ ∗

(2.66)

k,l=1

L′0

22

Alberto Barchielli

L′0 [X]

∞

1X ∗ := i[H, X] − Rk [Rk , X] + [X, Rk∗ ]Rk , 2

(2.67a)

k=1

Rk [X] := Skl [X] :=

∞ X

l=1 ∞ X j=1

Slk∗ [X, Rl ]

(2.67b)

∗ Sjk XSjl − δkl X .

(2.67c)

The master equation ρ(t)

The reduced statistical operator is defined by the partial trace ρ(t) := TrΓ {U (t)sU (t)∗ }

(2.68)

or, equivalently, by TrH {Xρ(t)} = TrH⊗Γ {jt (X)s} , ρ(f ; t)

∀X ∈ B(H).

If we set ρ(f ; t) := TrΓ {U (t) (ρ0 ⊗ η(f )) U (t)∗ } ,

(2.69)

then the reduced statistical operator, for the three choices discussed in Section 2.3, turns out to be given by case S1 ρ(t) = ρ(0; t); case S2 ρ(t) = ρ(f; t); case S3 ρ(t) = Ec ρ(f ; t) . We can always write

ρ(t) = Ec ρ(f ; t)

(2.70) c

by understanding that the classical expectation E has no effect when f is not random. By Propositions 2.5 and 2.2, ρ(f ; t) satisfies the integral equation ρ(f ; t) = ρ0 +

Z

0

L(f ), Rk (f ), H(f )

t

ds L f (s) [ρ(f ; s)],

(2.71)

where the integral can be interpreted again in the topology induced by the duality with B(H) and L f (s) is the bounded operator on T (H) given by ∞

1X ([Rk (f )̺, Rk (f )∗ ] + [Rk (f ), ̺Rk (f )∗ ]) , 2 k=1 (2.72a) Rk (f ) = C(zk )∗ [R + (S − 1l) C(f )] , (2.72b)

L(f )[̺] = −i [H(f ), ̺] +


ih H(f ) = H + i [C(f )∗ R − R∗ C(f )] + C(f )∗ (S − S ∗ ) C(f ) 2

i − R∗ (S − 1l) C(f ) + C(f )∗ (S ∗ − 1l) R .

23

(2.72c)

For a sufficiently regular f , eq. (2.71) can be differentiated and becomes the quantum master equation d ρ(f ; t) = L f (t) [ρ(f ; t)]. (2.73) dt The infinitesimal generator L f (t) is known as Liouville operator or Liouvillian; the name comes from the Liouville equation in classical statistical mechanics, which became in quantum mechanics the Liouville–von Neumann equation dρ(t)/dt = −i[H, ρ(t)] for an isolated system and the master equation (2.73) for an open system without memory. From (2.72a) the Lindblad structure ([84] Corollary 30.13 p. 268) of the Liouvillian is apparent; by (2.36), a more compact form is L(f )[̺] = −

1 ∗ 1 ∗ R R + R∗ SC(f ) + iH ̺ − ̺ R R + C(f )∗ S ∗ R − iH 2 2 ∗ + TrZ R + SC(f ) ̺ R + C(f )∗ S ∗ − kf k2 ̺ . (2.74)

In the case S1 one has the autonomous master equation d ρ(t) = L0 [ρ(t)], dt

(2.75)

where L0 ≡ L(0) is given by eq. (2.72a) with f = 0 and its adjoint L′0 is given in eq. (2.67a). In the case S2 the reduced operator ρ(t) satisfies the master equation (2.73), with a time dependent Liouvillian. Finally, in the case S3, the reduced operator does not satisfy any closed equation. 2.5 Physical basis of the use of QSC The quasi–monochromatic paraxial approximation of the electromagnetic field In 1978 Yuen and Shapiro [89] started to develop a theory of quantum–field propagation as a boundary–value problem; in particular, they treated the case of quasi–monochromatic paraxial fields, i.e. fields whose significant (nonvacuum state) modes all have temporal frequencies in the vicinity of a nominal carrier frequency ω0 and satisfy k ⊥ ≪ (ω0 /c)2 , where k ⊥ is the wave–vector component orthogonal to the direction of propagation. In such approximations they found, for the positive and negative frequency parts of the electric field, the CCR with a Dirac delta in time, as in (2.27). Moreover, they showed that, in the quasimonochromatic paraxial limit, the spatial propagation along

24

Alberto Barchielli

the axial direction involves pure translation in time, as for the solutions of the one–dimensional wave equation. These ideas were used by Frigerio and Ruzzier in an attempt to develop a relativistic version of QSC [52] and by the author in developing a photon detection theory [18]. A consequence of the idea of the quasimonochromatic paraxial approximation is that the Hilbert space Z, which appear in the definition of the Fock space Γ , must contain the field degrees of freedom linked to the nominal carrier frequencies involved in the problem, the directions of propagation, the polarization. The possible polarization states for the electromagnetic field are only two (two linear or two circular polarizations) and often, for sake of simplicity, polarization is even not considered. The other two things, nominal carrier frequencies and directions of propagation, depend on what is relevant in the matter–field interaction and, so, the choice of Z depend on the specific problem and, in particular, on the quantum system SH . Approximations in the system–field interaction In Ref. 54 Gardiner and Collet discuss the physical approximations needed to pass from a quasi–physical Hamiltonian to an evolution like (2.38), at least in the case S = 1l. They work with the Heisenberg equations of motion for system operators and obtain at the end equations like (2.66); let us give an idea of these approximations by working with the evolution operator [18]. Let us consider our system SH interacting with some Bose fields aj (ω) (we are in the frequency domain), satisfying the CCR’s [ai (ω), aj (ω ′ )] = 0,

[ai (ω), a†j (ω ′ )] = δij δ(ω − ω ′ ) .

(2.76)

A generic system–field interaction, linear in the field operators and in the rotating wave approximation, can be written as X 1 Z Ωj +θj √ HI = i kj (ω) Rj a†j (ω) − Rj∗ aj (ω) dω , (2.77) 2π Ωj −θj j where the Rj are system operators and the kj (ω) are real coupling constants. We shall consider the limiting case in which the coupling constants become independent of frequency and the field spectrum becomes flat and broad [kj (ω) → 1, θj → +∞]. In quantum optics this is often a good approximation [54]. First we pass to the interaction picture with respect to the free dynamics of the fields and take kj (ω) = 1 (flat spectrum). The interaction Hamiltonian becomes i Xh Rj a ˜†j (t) − Rj∗ a ˜j (t) , (2.78) HI (t) = i j

1 a ˜j (t) := √ 2π

Z

Ωj +θj

Ωj −θj

aj (ω) e−iωt dω .

(2.79)


25

˜t in the interaction picture can be Moreover, the time evolution operator U written as n Z t o − ˜ (t) = ← U T exp −i [H + HI (s)] ds , (2.80) 0

← − where H is the system Hamiltonian and T is the time–ordering prescription, which is usual in theoretical physics; for a mathematical presentation of the time ordered exponentials in QSC see Ref. 63. By taking the limit θj → +∞ (broad–band approximation), we obtain for˜j (t) satisfy the CCR (2.27). Moremally that the quantities aj (t) = lim a θj →+∞

over, we have Z

−iHI (s) ds

t

θj →+∞

a ˜j (s) ds −→ Aj (t) , 0 h i X θj →+∞ Rj dA†j (s) − Rj∗ dAj (s) , −→

(2.81a) (2.81b)

j

θj →+∞

˜ (t) U

−→ U (t) , Z th i X ← − † ∗ Rj dAj (s) − Rj dAj (s) H ds + i U (t) ≡ T exp −i . 0

(2.81c) (2.82)

j

By writing U (t + dt) in the form n io Xh Rj dA†j (t) − Rj∗ dAj (t) Ut , U (t + dt) = exp −iH dt +

(2.83)

j

by expanding the exponential and by using the multiplication table (2.31), one sees that U (t) satisfies the Hudson–Parthasarathy equation (2.38) with S = 1l. Note that the term − 21 Rj† Rj dt in K (2.36e) comes from the second order term in the expansion of the exponential. Let us stress that this construction shows that the fields aj (t) used in QSC are the formal limit θj → +∞ of the quantities (2.79). This explains once again the fact that a field aj (t) has to be considered as a wave packet with some carrier frequency Ωj and bandwidth 2θj ; then, the approximation of infinite bandwidth is taken. Many other limiting schemes justifying the Hudson–Parthasarathy equation have been developed, also in mathematically rigourous forms [4–6].

3 Continual measurements 3.1 Field observables and indirect measurements on SH When the fields represent pure noise, it is natural to consider system observables as in Section 2.4; but in other situations, as when the fields are intended

26

Alberto Barchielli

to represent the electromagnetic field, the natural observables are field quantities, from which inferences are done on the system SH . We are interested in the behaviour of the system SH , but we measure field observables; this scheme is known as indirect measurement. Another way to think to this situation is the following one. We cannot act directly on our system SH , but any action is mediated by some quantum input and output channel. We can think of an atom driven by a laser (input) and emitting fluorescence light (output) or of the light entering (input) and leaving (output) an optical cavity. In these examples the role of input and output channels is played by the electromagnetic field and we can think of approximating it by the Bose fields on which QSC is based. So, we have to identify the main field observables, which eventually we want to take under measurement with continuity in time. Counts of quanta N (P ; t)

Let P ∈ B(Z) be an orthogonal projection; for any t ≥ 0, we introduce the operator X N (P ; t) := λ P ⊗ 1(0,t) = hzk |P zl iΛkl (t) . (3.1) kl

By Theorem 2.1, this operator is essentially selfadjoint on E and its domain includes also the finite particle number vectors. From (2.24c) we have he(g)|N (P ; t)e(f )i = exp hg(0,t) |(1l − P )f(0,t) i + hg(t |f(t i ∞ X n n × hg(0,t) |P f(0,t) i ; (3.2) n! n=0

by taking into account the factorization (2.9), one sees that the eigenvalues of N (P ; t) are the integers n = 0, 1, . . . and that the eigenspace corre sponding to n is the “n-particle sector of Γsymm (P Z) ⊗ 1(0,t) L2 (R+ ) ” ⊗ Γsymm 1l − P ⊗ 1(0,t) Z ⊗ L2 (R+ ) . Therefore, we can interprete N (P ; t) as the number operator which counts the quanta injected in the system up to time t with state in P Z. Another way to see that N (P ; t) is a number operator is to use the heuristic rules of QSC; by (3.1), (2.31) and the fact that P is a projection, we have immediately 2 dN (P ; t) = dN (P ; t) , (3.3) which shows that an infinitesimal increment has eigenvalues 0 and 1. By (2.16) and (3.1), we have exp{iκN (P ; t)} = W 0; exp{iκP ⊗ 1(0,t) }

(3.4)

and by (2.13) one sees that the unitary groups generated by N (P ; t) and N (P ; s) commute; therefore, {N (P ; t), t ≥ 0} is a set of jointly diagonalizable


27

selfadjoint operators, or, in physical terms, of compatible observables. The same is true for N (Pα ; t) t ≥ 0 , α = 1, 2, . . . with Pα Pβ = δαβ Pα = δαβ Pα∗ , (3.5)

i.e. P1 , P2 , . . . are mutually orthogonal projections. In the case of photons the measurement of number operators can be experimentally realized through the so called direct detection, which we present in Section 3.4. Measurements of field quadratures Let us consider now the field quadratures Z t Z t X † Q(h; t) := Q h(0,t) = hk (s) dAk (s) + hk (s) dAk (s) , k

0

Q(h; t) (3.6)

0

which are essentially selfadjoint operators on √ E (Theorem 2.1). The√spectrum of Q(h; t) is the whole real axis because ( 2 khk)−1 Q(h; t) and ( 2 khk)−1 Q(ih; t) form a couple of canonically conjugated selfadjoint operators (the commutator gives i). By (2.14), (2.15), we have that 2 Q(hα ; t) , t ≥ 0 , α = 1, 2, . . . , with hhα (s)|hβ (s)i = δαβ khα (s)k , (3.7) is a family of compatible observables. Definition 2.2 and Proposition 2.1 guarantee that the quantum stochastic integral in (3.6) is well defined In the case of photons the measurement of field quadratures can be experimentally realized through the so called heterodyne detection, which we present in Section 3.5. Field observables in the Heisenberg picture In Section 2.2 we discussed the fact that all the fields we have introduced are expressed in the interaction picture and, so, this is true also for the field observables (3.1) and (3.6). However, in order to construct a theory of continual measurements, based on the usual rules of quantum mechanics, which require the existence of joint spectral measures, we need observables commuting at different times in the Heisenberg picture; so, we have to show that the observables introduced above continue to commute at different times even in the Heisenberg picture. Let us call “input fields” the fields before the interaction with the system SH , i.e. the fields Ak (t), A†k (t), Λkl (t), . . . and let us call “output fields” the fields after the interaction with the system SH or, in other words, the fields in the Heisenberg picture. We have ∗ Aout j (t) := U (t) Aj (t)U (t)

(3.8)

Aout j (t)

28 † Aout (t), Λout ij (t), j

Qout (h; t), N out (P ; t)

Alberto Barchielli

† out and similar definitions for Aout (t), Λout (h; t), N out (P ; t). Note that, ij (t), Q j ∗ if D is the domain of Aj (t), then U (t) D is the domain of Aout j (t) and similar statements for the other operators; Qout (h; t), N out (P ; t) remain selfadjoint operators. By Theorem 2.3 we have

U (T ) = U (T, t)U (t) ,

∀T ≥ t ,

(3.9)

with U (T, t) adapted to H ⊗ Γ(t,T ) and, so, commuting with Aj (t), A†j (t), Λij (t), . . . Therefore, we have ∗ Aout j (t) = U (T ) Aj (t)U (T ) ,

∀T ≥ t .

(3.10)

This implies immediately that the output fields satisfies the same commutation rules of the input fields, for instance the CCR (2.25): the output fields remain Bose free fields. Moreover, we have that now {N out (Pj ; t), t ≥ 0, j = 1, 2, . . .}, with P1 , P2 , . . . mutually othogonal projections, and {Qout (hj ; t), t ≥ 0, j = 1, 2, . . .}, with h1 (s), h2 (s), . . . mutually orthogonal vectors for any s, are two families of compatible observables in the Heisenberg picture as we wanted. By applying the formal rules of QSC we can express the output fields as the quantum stochastic integrals [14] Z t X ∗ ∗ out U (s) Sjk U (s)dAk (s) + U (s) Rj U (s) ds , (3.11a) A j (t) = 0

† Aout (t) = j

Λout ij (t) =

k

Z t X 0

k

Z t X 0

kl

+

X l

∗ U (s)∗ Sjk U (s)dA†k (s) + U (s)∗ Rj∗ U (s) ds ,

∗ Sjl U (s) dΛkl (s) + U (s)∗ Sik

X

(3.11b)

∗ Rj U (s) dA†k (s) U (s)∗ Sik

k

U (s)∗ Ri∗ Sjl U (s) dAl (s) + U (s)∗ Ri∗ Rj U (s) ds .

(3.11c)

From these equations one explicitly sees that the output fields carry information on system SH : the quantities Rk , Skl are the system operators appearing in the system–field interaction. Moreover, these equations allow to write our observables as


N out (P ; t) =

Z t X 0

kl

+

X

U (s)∗ C(zk )∗ S ∗ (1l ⊗ P ) SC(zl )U (s) dΛkl (s)

U (s)∗ C(zk )∗ S ∗ (1l ⊗ P ) RU (s) dA†k (s)

k

+

X l

U (s)∗ R∗ (1l ⊗ P ) SC(zl )U (s) dAl (s) + U (s) R (1l ⊗ P ) RU (s) ds , ∗

out

Q

(h; t) =

Z t X 0

k

+

29

(3.12)

U (s)∗ C(zk )∗ S ∗ C h(s) U (s) dA†k (s)

X l

∗

∗ U (s)∗ C h(s) SC(zl )U (s) dAl (s)

h ∗ i + U (s)∗ R∗ C h(s) + C h(s) R U (s) ds ;

(3.13)

these two equations give us an idea of how the output observables depend on the field operators and on the system ones. In eqs. (3.11)–(3.13) it is possible to check that the quantum stochastic integrals in the r.h.s. are all well defined, but a rigorous proof of anyone of these formulae needs to use two times Proposition 2.4 and to control the domain of the triple product. However we do not need this, because we can shortcut this difficulty by using Weyl operators. 3.2 Characteristic functionals Let us recall some very well known facts about selfadjoint operators (observables) and their distribution laws; see for instance Section 10 of Ref. 84. Let X be a selfadjoint operator and exp {ikX} the group generated by X; then, there exists a unique projection–valued measure (pvm) ξ on R, B(R) such that Z ikX e = eikx ξ(dx) , ∀k ∈ R . (3.14) R

Let X ≡ (X1 , . . . , Xd ) be a set of d mutually commuting selfadjoint operators, in the sense that the groups generated by them commute or that theassociated pvm ξj commute; then, there exists a unique pvm ξ on Rd , B(Rd ) such that eik·X ≡

d Y

j=1

eikj Xj =

Z

Rd

eik·x ξ(dx) ,

∀k ∈ Rd .

(3.15)

Moreover, in the state ̺, the characteristic function (Fourier transform) of the probability law

pvm

30

Alberto Barchielli

P̺X (dx) = Tr ̺ ξ(dx)

(3.16)

of the observable associated to X is Z eik·x P̺X (dx) = Tr ̺ eik·X .

(3.17)

Rd

These results extend to “infinitely many” commuting selfadjoint operators; only uniqueness is lost. Proposition 3.1. Let {ξt , t ∈ T } be a family of commuting pvm on R, B(R) and let Xt be the selfadjoint operator associated with ξt . Then, there exist a measurable space n (Ω, F), aopvm ξ on (Ω, F) and a family of real valued ˜ t , t ∈ T on Ω such that measurable functions X eikXt ≡

Z

eikx ξt (dx) =

Z

˜

eikXt (ω) ξ(dω) ,

Ω

R

∀k ∈ R, t ∈ T .

(3.18)

Proof. This proposition is given as an exercise by Parthasarathy ([84] Exercise 10.11 p. 59), with the hint: use Bochner’s Theorem, Kolmogorov’s Consistency Theorem and Zorn’s Lemma. ⊓ ⊔ In the situation described in this proposition, if ̺ is a fixed state and we set P̺ (dω) := Tr{̺ ξ(dω)} ,

o n ˜ t (·) , t ∈ T bewe have that (Ω, F, P̺ ) is a classical probability space and X comes a classical stochastic process. The characteristic functions of the finite– dimensional distributions of this process are given by X X Z n n ˜ t (ω) P̺ (dω) = Tr ̺ exp i exp i kj X . (3.19) k X j tj j Ω

j=1

j=1

Let us consider now the case in which the index becomes time plus a discrete label: T = {(α, s) : α = 1, . . . , d, 0 < s ≤ t}. Then, we denote the ˜ process by X(α, s; ω), the operators by X(α, s) and we assume, for simplicity, ˜ X(α, 0; ω) = 0, X(α, 0) = 0. Instead of considering the finite–dimensional ˜ distributions of the process X(α, s), it is equivalent and simpler to introduce the finite–dimensional distributions of the increments of the original process, whose characteristic functions are Z

Ω

X d X n ˜ ˜ exp i kα (sj ) X(α, sj ; ω) − X(α, sj−1 ; ω) P̺ (dω) α=1 j=1

X d X n = Tr ̺ exp i , kα (sj ) X(α, sj ) − X(α, sj−1 ) α=1 j=1

0 = s0 < s1 < · · · < sn ≤ t .

(3.20)


31

The characteristic functional for the counts of quanta Let us consider the family of compatible observables {N (Pα ; t), t ≥ 0, α = 1, 2, . . . , d}; P1 , . . . , Pd are mutually othogonal projections on Z. According to the discussion above and Proposition 3.1, we can handle the stochastic process associated to these operators by means of the finite–dimensional characteristic functions for the increments, which in turn can be summarized in a characteristic functional, which is suggested by the structure of eq. (3.20) and which now we construct. Let us introduce the test functions k ∈ L∞ R+ ; Rd , the unitary operators St (k) on L2 (R+ ; Z) by X d St (k) := exp i Pα ⊗ 1(0,t) kα

(3.21a)

α=1

or by d X kα (s)Pα f (s) St (k)f (s) = exp i1(0,t) (s) α=1

≡ 1(0,t) (s)

d h X

α=1

i eikα (s) − 1 Pα f (s) + f (s)

(3.21b)

and the characteristic operator bt (k) := W 0; St (k) . Φ

(3.22)

bt (κk) is a unitary group with selfadjoint By (2.16) and (3.21a) the map κ 7→ Φ generator λ

X α

Pα ⊗ 1(0,t) kα

XZ t X zk = kα (s)Pα zl dΛkl (s) kl

0

α

≡

XZ α

t

kα (s) dN (Pα ; s)

(3.23)

0

and, so, we can write XZ t b kα (s) dN (Pα ; s) . Φt (k) = exp i α

(3.24)

0

bt (k) Now, by recalling that our initial state is s and that N (Pα ; s) and Φ contain the free–field dynamics, but not the system–field interaction, we can define the characteristic functional by n o bt (k)U (t)sU (t)∗ . Φt (k) := Tr Φ (3.25)

32

˜α (s) N

Alberto Barchielli

By taking in Φt (k) a simple function k, we obtain a characteristic function ˜α (s). of the type (3.20) for the increments of a process which we denote by N ˜ All the finite–dimensional distributions for the increments of Nα (s) and for ˜α (s) itself are contained in the characteristic functional (3.25). By using the N output fields and in particular the property (3.10), it is suggestive to write ( ) XZ t kα (s) dN out (Pα ; s) s , (3.26) Φt (k) = Tr exp i α

0

where Heisenberg–picture commuting operators N out (Pα ; s) explicitly appear. The characteristic functional for the field quadratures Let us consider the family of compatible observables {Q(hα ; t), t ≥ 0, α = 2 1, 2, . . . , d}, with hhα (s)|hβ (s)i = δαβ khα (s)k ; we can repeat the construction of the previous subsection. By (3.6) we have d Z X

α=1

0

t

X kα (s) dQ(hα ; s) = Q kα hα ; t

(3.27)

α

and, by taking into account (2.15), we can write the characteristic operator as an adapted Weyl operator again: X d Z t b Φt (k) = exp i kα (s) dQ(hα ; s) α=1

0

X X kα hα 1(0,t) ; 1l . =W i kα hα ; t = exp iQ

˜ α (s) Q

(3.28)

α

α

˜ α (s) associated with the Then, the characteristic functional of the process Q selfadjoint operators Q(hα ; s) is given by (3.25) again or by ( ) XZ t out kα (s) dQ (hα ; s) s . (3.29) Φt (k) = Tr exp i α

0

Field observables and adapted Weyl operators By the use of adapted Weyl operators and characteristic operators we can unify the field observables (3.1) and (3.6) and generalize them by including observables of the type (2.17). By putting together the structures (2.18) and bt (k) Φ bt (k ′ ) = Φ bt (k + k ′ ), which gives the commuta(3.28) and by imposing Φ tivity of the associated observables, we arrive to the following theorem.


33

1 2 d Theorem 3.1. Let h B α, B , . . .β,iB be commuting selfadjoint operators in Z, i.e. B α = B α∗ , eiκα B , eiκβ B = 0, ∀κα , κβ ∈ R, ∀α, β = 1, . . . , d. Let us

take c ∈ L1loc (R+ ; Rd ), b ∈ L2loc (R+ ; Z) and hα ∈ L2loc (R+ ; Z), α = 1, . . . , d, such that α

Imhhα (t)|hβ (t)i = 0 ,

eiκB hβ (t) = hβ (t) , ∀t ≥ 0,

∀κ ∈ R,

d Y α eikα (s)B , S k(s) =

(3.31b)

α=1

rt (k)(s) = 1(0,t) (s) r(k; s) , r(k; s) = i

α=1

(3.31c)

kα (s)hα (s) + S k(s) − 1l b(s) ,

(3.31d)

Z t X

bt (k) = exp i ds Φ kα (s)cα (s) + Im b(s) S k(s) b(s) 0

α

× W rt (k); St (k) .

(3.32)

Then, the characteristic operator has the following properties: 1. localization properties: bt 1(t ,t ) k = Φ bt 1(t ,t ) k ∈ U Γ(t ,t ) , Φ 2 1 2 1 2 1 2

0 ≤ t 1 < t2 ≤ t ;

(3.33)

2. group property:

bt (k) Φ bt (k ′ ) = Φ bt (k + k ′ ) , Φ

∀k, k ′ ∈ L∞ (R+ ; Rd ) ;

(3.34)

bt (k) is strongly continuous in k ∈ L∞ (R+ ; Rd ) and in t ≥ 0; 3. continuity: Φ 4. matrix elements: bt (k)e(f )i = he(g)|e(f )i he(g)|Φ Z t 1X ds − × exp kα (s)hhα (s)|hβ (s)ikβ (s) 2 0 αβ X α kα (s) c (s) + hhα (s)|f (s)i + hg(s)|hα (s)i +i α

+ g(s) + b(s) S k(s) − 1l f (s) + b(s)

c b hα

∀α, β = 1, . . . , d.

(3.30) For any test function k ∈ L∞ (R+ ; Rd ) let us define St (k) ∈ U L2 (R+ ; Z) , bt (k) ∈ U(Γ ) by rt (k) ∈ L2 (R+ ; Z) and the characteristic operator Φ St (k)f (s) = 1(0,t) (s) S k(s) − 1l f (s) + f (s) , ∀f ∈ L2 (R+ ; Z) , (3.31a)

d X

B1, B2, . . . , Bd

;

(3.35)

St (k) rt (k) bt (k) Φ

34

Alberto Barchielli

b0 (k) = 1l, Φ bt (k) is the unique unitary solution 5. given the initial condition Φ of the QSDE X

b b dΦt (k) = Φt (k) zl S k(t) − 1l zm dΛlm (t) +

X l

lm

hzl |r(k; t)idA†l (t) −

X l

hr(k; t)|S k(t) zl idAl (t)

X kα (t)cα (t) + hb(t)| S k(t) − 1l b(t)i + i α

1X α β − kα (t)hh (t)|h (t)ikβ (t) dt . 2

(3.36)

αβ

ξ(dω) ˜ X(α, t; ·) X(α, t)

Moreover, there exists a measurable space a n (Ω, F), a pvm ξ on (Ω, F), o ˜ family of real valued measurable functions X(α, t; ·) , α = 1, . . . , d, t ≥ 0 on Ω, a family of commuting and adapted selfadjoint operators X(α, t), α = ˜ 1, . . . , d, t ≥ 0 such that X(α, 0; ω) = 0, X(α, 0) = 0 and, for any choice of n, 0 = t0 < t1 < · · · < tn ≤ t, κjα ∈ R, X d n X j b κα X(α, tj ) − X(α, tj−1 ) Φt (k) = exp i j=1 α=1

=

Z

Ω

where kα (s) =

X n X d ˜ ˜ exp i κjα X(α, tj ; ω) − X(α, tj−1 ; ω) ξ(dω) ,

(3.37)

j=1 α=1

Pn

j=1

1(tj−1 ,tj ) (s) κjα .

Proof. Equations (3.33) follow immediately from the definition of the characteristic operator and from the properties of the Weyl operators. One can check that the definitions of St (k) and rt (k) are such that St (k)St (k ′ ) = St (k + k ′ ) , St (k)−1 = St (k)∗ = St (−k) , rt (k) + St (k)rt (k ′ ) = rt (k + k ′ ) .

(3.38) (3.39)

Together with (2.13) and (3.30), these equations imply (3.34). The matrix elements (3.35) follow by simple computations from the definition (2.10) of the Weyl operators. bt (k) and the fact that E is dense, it is enough to By the unitarity of Φ prove the strong continuity on the exponential vectors. By the unitarity and the properties (3.33) and (3.34), the strong continuity on the exponential vectors reduces to the check of the continuity of the matrix elements (3.35). By checking that the condition of Definition 2.2 is satisfied, one has that the r.h.s. of (3.36) is well defined. By differentiating the matrix elements (3.35)


35

one gets the QSDE (3.36); by passing to the equation for the matrix elements, which turns out to be a closed ordinary differential equation, one obtains the uniqueness of the solution. The last statement is an application of Proposition 3.1 to the present case. ⊓ ⊔ Continual measurements and infinitely divisible laws ˜ It is important to realize that in a coherent state ψ(f ) the process X(α, t) has independent increments; here we are not considering the interaction with system SH . Indeed, by (3.35), (3.37) and (2.7), one obtains bt (k)ψ(f )i = hψ(f )|Φ

n D Y

j=1

X d ψ(1(tj−1 ,tj ) f ) exp i κjα X(α, tj ) α=1

E − X(α, tj−1 ) ψ(1(tj−1 ,tj ) f ) ;

(3.40)

by the localization properties (3.33), we can reintroduce ψ(f ) in every factor and we obtain, again by (3.37), the independence of the increments: Z

Ω

X n X d j ˜ ˜ exp i κα X(α, tj ; ω) − X(α, tj−1 ; ω) hψ(f )|ξ(dω)ψ(f )i

=

j=1 α=1

n Z Y

j=1

Ω

X d j ˜ ˜ κα X(α, tj ; ω) − X(α, tj−1 ; ω) hψ(f )|ξ(dω)ψ(f )i. exp i α=1

(3.41)

This fact implies that, in a time–homogeneous case, the increments follow an infinitely divisible law. Indeed, let us take f = 0, b(s) = b, c(s) = c, h(s) = h, kα (s) = κα and let us denote by ζB (dx) the joint spectral measure of the B’s; then, one has

X d Y E D bt (κ) ψ(0) ≡ ψ(0) κα cα eiκα X(α,t) ψ(0) = exp it ψ(0) Φ α=1

α

Z h X i tX − exp i κα hhα |hβ iκβ + t κα xα − 1 hb|ζB (dx)bi . 2 Rd α

(3.42)

αβ

By comparing this result with the Lévy–Khintchin formula, one sees that (3.42) is the characteristic function of an infinitely divisible distribution on Rd , but not the most general one because hb|ζB (dx)bi is a finite measure. The representations of infinitely divisible laws have been studied ([84] Section 21) and from there we can take a suggestion on how we can generalbt (k) are mainize the observables of Theorem 3.1. All the properties of Φ tained and the most general infinitely divisible distribution is obtained if

36

Alberto Barchielli

q 1+|B|2 everywhere in Theorem 3.1 b(s) is replaced by |B|2 b(s) and cα (s) by 1+|B|2 α

Pd 2 α 2 cα (s) − b(s) |B|2 B b(s) ; here |B| := α=1 (B ) . This is only an indication of a possible generalization, but we do not touch any more this more general characteristic operator in this work. The output characteristic operator bout (k) Φ t

The observables in the Heisenberg description can now be introduced implicitly by defining the output characteristic operator

Formally we have

∗b bout Φ t (k) = U (t) Φt (k)U (t) .

XZ t out bout (k) = exp i Φ k (s) dX (α; s) , α t α

(3.43)

(3.44)

0

but we do not need to give a rigourous meaning to the “Heisenberg” observables X out (α; t) and to the integrals with respect to dX out (α; s); all we need bout is to differentiate Φ t (k), which is done in Proposition 3.2.

The characteristic functional and the finite dimensional laws Φt (k)

∆X(t1 , t2 ) ξ(dx; t1 , t2 )

By considering also the interaction with system SH , we have that the charac˜ is again given by (3.25): teristic functional of the process X o n o n bt (k)U (t)sU (t)∗ ≡ Tr Φ bout (k)s . (3.45) Φt (k) = Tr Φ t

All the probabilities describing the continual measurement of the observables X(α, t) are contained in Φt (k); let us give explicitly the construction of the joint probabilities for a finite number of increments. o n ˜ The measurable functions X(α, t; ·) , α = 1, . . . , d, t ≥ 0 , introduced in Theorem 3.1, represent the output signal of the continual measurements. Let ˜ ˜ ˜ ˜ us denote by ∆X(t1 , t2 ) = X(1, t2 ) − X(1, t1 ), . . . , X(d, t2 ) − X(d, t1 ) the

vector of the increments of the output in the time interval (t1 , t2 ) and by ξ(dx; t1 , t2 ) the joint pvm on Rd of the increments X(α; t2 ) − X(α; t1 ), α = 1, . . . , d. Note that, because of the properties of the characteristic operator, not only the different components of an increment are commuting, but also increments related to different time intervals; this implies that the pvm related to different time intervals commute. Moreover, the localization properties of the characteristic operator give ξ(A; t1 , t2 ) ∈ B(Γ(t1 ,t2 ) ) ,

for any Borel set A ⊂ Rd .

(3.46)


37

As in the last part of Theorem 3.1, letus consider 0 = t0 < t1 < · · · < Pn tn ≤ t, kα (s) = j=1 1(tj−1 ,tj ) (s) κjα , κj = κj1 , . . . , κjd ; then we can write X d n X j ∗ κα X(α, tj ) − X(α, tj−1 ) U (t)sU (t) Φt (k) = Tr exp i =

Z

Rnd

Y n

j=1

j=1 α=1

e

iκj ·xj

Pρ0 ∆X(t0 , t1 ) ∈ dx1 , . . . , ∆X(tn−1 , tn ) ∈ dxn ,

(3.47)

where the physical probabilities are given by Pρ0 ∆X(t0 , t1 ) ∈ A1 , . . . , ∆X(tn−1 , tn ) ∈ An Y n ξ(Aj ; tj−1 , tj ) U (t)sU (t)∗ . = Tr

(3.48)

j=1

Obviously, Φt (k) is the characteristic function of the physical probabilities Pρ0 ∆X(t0 , t1 ) ∈ A1 , . . . , ∆X(tn−1 , tn ) ∈ An and it uniquely determines them. 3.3 The reduced description of the continual measurements An essential step in the theory is to eliminate the degrees of freedom of the fields and to pass to a reduced description based only on system SH . This is similar to the passage to the reduced dynamics in Section 2.4 and, indeed, the quantities ρ(f ; t) (2.69) and L f (t) (2.72) will be involved also here. The reduced characteristic operator

Definition 3.1. For f ∈ L2 (R+ ; Z), k ∈ L∞ (R+ ; Rd ), t ≥ 0, let us define the reduced characteristic operator Gt (f ; k) ∈ B T (H) by: ∀a ∈ B(H), ∀̺ ∈ T (H), n o bt (k) U (t) ̺ ⊗ η(f ) U (t)∗ . TrH a Gt (f ; k)[̺] = TrH⊗Γ a ⊗ Φ (3.49)

Let us recall that the symbol η(f ) for a coherent state is defined in eq. (2.57). By the definition of reduced characteristic functional, for the three choices of initial state proposed in Section 2.3 the characteristic functional is given by S1. Φt (k) = TrH Gt (0; k)[ρ0 ] , S2. Φt (k) = TrH Gt (f ; k)[ρ0 ] , S3. Φt (k) = Ec TrH Gt (f ; k)[ρ0 ] ; let us recall that in this case f is random.

Gt (f ; k)

38

Alberto Barchielli

In (3.49) only unitary operators are involved and this implies that it is easy to apply Proposition 2.4 to the triple product; we get the following results. bout Proposition 3.2. Let Φ t (k) and Gt (f ; k) be defined by (3.43), (3.49), with all the hypotheses given in the previous sections. The output characteristic operator (3.43) satisfies the QSDE X out ∗ b dΦt (k) = U (t) C(zl )∗ S ∗ S k(t) − 1l SC(zm ) dΛlm (t) +

+

X

Xh l

l

lm

C(zl )∗ S ∗ C r(k; t) + S k(t) − 1l R dA†l (t)

∗ i −C r(k; t) S k(t) + R∗ S k(t) − 1l SC(zl ) dAl (t) X + hb(t)| S k(t) − 1l b(t)i + i kα (t)cα (t) α

1X − kα (t)hhα (t)|hβ (t)ikβ (t) + R∗ C r(k; t) 2 αβ ∗ bout − C r(k; t) S k(t) R + R∗ S k(t) − 1l R dt U (t)Φ t (k) .

The reduced characteristic operator satisfies the equation Z t Ks f ; k(s) ◦ Gs (f ; k)[̺] ds , Gt (f ; k)[̺] − ̺ =

(3.50)

(3.51)

0

Kt (f ; κ)

with generator

n o Kt (f ; κ)[̺] = L f (t) [̺] + TrZ 1l ⊗ (S(κ) − 1l) J (f ; t)[̺] X 1X κα Z α (f ; t)̺ + ̺Z α (f ; t)∗ , (3.52a) κα hhα (t)|hβ (t)iκβ ̺ + i − 2 α αβ

∗ 1 α c (t) + C hα (t) R + SC f (t) , (3.52b) 2 h ∗ ∗ i J (f ; t)[̺] = R + SC f (t) + C b(t) ̺ R∗ + C f (t) S ∗ + C b(t) ; (3.52c) L f (t) is given by eq. (2.74). Z α (f ; t) =

Z α (f ; t), J (f ; t)

bt (k); due to the unitarity Proof. Let us apply Proposition 2.4 to U (t)∗ a ⊗ Φ of the adapted processes involved, there is no problem to control the domains and we get X ∗ ∗b b d U (t) a ⊗ Φt (k) = U (t) Φt (k) C(zl )∗ S ∗ a ⊗ S k(t) − a ⊗ 1l lm


× C(zm ) dΛlm (t) − +

X l

X

C(zl )∗ S ∗

l

39

R − C r(k; t) a dA†l (t)

∗ a ⊗ S k(t) C(zl ) dAl (t) R − C r(k; t) ∗

X kα (t)cα (t) a + K ∗ a + hb(t)| S k(t) − 1l b(t)ia + i α

1X α β ∗ − kα (t)hh (t)|h (t)ikβ (t) a + R C r(k; t) a dt . 2

(3.53)

αβ

By appling Proposition 2.4 again we get X ∗ ∗ b d U (t) a ⊗ Φt (k) U (t) = U (t) C(zl )∗ S ∗ a ⊗ S k(t) S − a ⊗ 1l × C(zm ) dΛlm (t) − +

X h l

X

∗

C(zl ) S

l

∗

lm

R − C r(k; t) a − a ⊗ S k(t) R dA†l (t)

i ∗ R − C r(k; t) a ⊗ S k(t) − aR∗ SC(zl ) dAl (t) ∗

X + L′0 [a] + hb(t)| S k(t) − 1l b(t)ia + i kα (t)cα (t) a α

1X − kα (t)hhα (t)|hβ (t)ikβ (t) a + R∗ C r(k; t) a 2 αβ ∗ bt (k) U (t). − C r(k; t) a ⊗ S k(t) R + R∗ a ⊗ S k(t) − 1l R dt Φ

(3.54)

By taking the matrix elements with respect to the coherent vector |ψ(f )i we get eqs. (3.51), (3.52a); by taking a = 1l we get (3.50). ⊓ ⊔ If the time dependence of the generator is sufficiently regular we have the differential equation d Gt (f ; k)[̺] = Kt f ; k(t) ◦ Gt (f ; k)[̺] , G0 (f ; k) = 1l , (3.55) dt which is a modification of a master equation. This kind of equations was introduced in [22, 23], inside of the operational approach, and it was related to the approach based on QSDE in [24]. The problem of generators of the type (3.52a) was studied in [25, 61, 62]. Let us define a two–time modification of the reduced characteristic operator by Z Gtt12 (f ; κ)[̺] − ̺ =

t2

t1

Ks (f ; κ) ◦ Gts1 (f ; κ)[̺] ds .

Then, for 0 = t0 < t1 < · · · < tn ≤ t, kα (s) = κj1 , . . . , κjd , eq. (3.51) gives

Pn

j=1

(3.56)

1(tj−1 ,tj ) (s) κjα , κj =

Gtt12 (f ; κ)

40

Alberto Barchielli n Gt (f ; k) = Gttn (f ; 0) ◦ Gttn−1 (f ; κn ) ◦ · · · ◦ Gtt01 (f ; κ1 ) .

(3.57)

Moreover, we have Gtt23 (f ; κ) ◦ Gtt12 (f ; κ) = Gtt13 (f ; κ) .

(3.58)

The reduced dynamics Υ (f ; t, s)

Let us set Υ (f ; t, s) := Gst (f ; 0) ;

(3.59)

by (3.57) we have the composition law Υ (f ; t3 , t1 ) = Υ (f ; t3 , t2 ) ◦ Υ (f ; t2 , t1 ) ,

0 ≤ t 1 ≤ t2 ≤ t3 .

(3.60)

Then, by comparing (2.71) with (3.51), (3.52a), (3.60) we obtain ρ(f ; t) = Υ (f ; t, 0)[ρ0 ] = Υ (f ; t, s)[ρ(f ; s)] .

(3.61)

In the case S2 the reduced statistical operator is ρ(t) = ρ(f ; t) and, so, Υ (f ; t, s) has the meaning of reduced evolution operator from s to t; eq. (3.60) says that this evolution is in some sense without memory. In the case S3 we have ρ(t) = Ec Υ (f ; t, 0)[ρ0 ] = Ec Υ (f ; t, s)[ρ(f ; s)] (3.62) and the lack–of–memory property is lost. From eqs. (3.56), (3.59), (3.52a) we get Z t2 Υ (f ; t2 , t1 )[̺] − ̺ = L f (s) ◦ Υ (f ; s, t1 )[̺] ds ,

(3.63)

t1

which is another form of the master equation (2.73). Instruments and finite–dimensional laws

Itt12 (f ; A)

instrument

In the quantum theory of measurement an important notion is that of instrument and the operational approach to continual measurements, mentioned in the Introduction, is based on such a notion. Here we recall a few facts, without developing in full this side of the theory. By using the joint pvm ξ(dx; t1 , t2 ) of the increments X(α; t2 ) − X(α; t1 ), α = 1, . . . , d, we define the map–valued measure Itt12 (f ; ·), 0 ≤ t1 < t2 , f ∈ L2 (R+ ; Z), by: ∀a ∈ B(H), ∀̺ ∈ T (H), TrH a Itt12 (f ; A)[̺] = TrH⊗Γ {a ⊗ ξ(A; t1 , t2 ) U (t2 , t1 ) ̺ ⊗ η(f ) U (t2 , t1 )∗ } , (3.64) where A is a Borel set in Rd ; by the factorization properties of Γ and η(f ), only f(t1 ,t2 ) , the part of f in (t1 , t2 ), is relevant for the definition of Itt12 (f ; A). The family of maps Itt12 (f ; ·) is a completely positive instrument [44,64], whose characterizing properties are


41

1. Itt12 (f ; A) ∈ B T (H) ; t2 2. Tr It1 (f ; Rd )[̺] = Tr {̺}, ∀̺ ∈ T (H); n X 3. Tr a∗i aj Itt12 (f ; A) |ψj ihψi | ≥ 0, ∀n, ∀ψj ∈ H, ∀aj ∈ B(H); i,j=1

4. for any finite or countable (Borel) partition A1 , A2 , . . . of a Borel set A P one has j Itt12 (f ; Aj )[̺] = Itt12 (f ; A)[̺], ∀̺ ∈ T (H).

Then, Gtt12 (f ; κ) is the Fourier transform of the instrument Itt12 (f ; ·), i.e. Z t2 Gt1 (f ; κ)[̺] = eiκ·x Itt12 (f ; dx)[̺] , ∀̺ ∈ T (H) , (3.65) Rd

and a Bochner type theorem holds which says that G uniquely determines I, see [25] p. 110 Theorem 1.5. By (3.47), (3.48), (3.57), (3.65) the probabilities (3.48) are given by Pρ0 ∆X(t0 , t1 ) ∈ A1 , . . . , ∆X(tn−1 , tn ) ∈ An oi h n n = Ec TrH Ittn−1 (f ; An ) ◦ · · · ◦ Itt01 (f ; A1 )[ρ0 ] ;

(3.66)

we use the convention that the classical expectation Ec has no effect when f is not random. Mean values and covariances ˜ As usual, all the moments of the process X(α, t) can be obtained by derivation of the characteristic functional. If we take kα (t) = κ1 δαα1 1(0,t1 ) (t) + κ2 δαα2 1(0,t2 ) (t) , the mean and the second moments are given by i h ˜ j , tj ) = −i ∂ Φt ∨t (k) Eρ0 X(α , 1 2 ∂κj κ=0

i h 2 ˜ 1 , t1 )X(α ˜ 2 , t2 ) = − ∂ , Φt1 ∨t2 (k) Eρ0 X(α ∂κ1 ∂κ2 κ=0 when they exist. Obviously, the covariance function is

i h ˜ 1 , t1 ), X(α ˜ 2 , t2 ) Cov X(α i h i h i h ˜ 2 , t2 ) . ˜ 1 , t1 ) Eρ X(α ˜ 1 , t1 )X(α ˜ 2 , t2 ) − Eρ X(α = Eρ0 X(α 0 0

(3.67)

(3.68) (3.69)

(3.70)

Proposition 3.3. Let the operators B α be bounded; then we have Z tj ∂ −i = TrH {Gt1 ∨t2 (f ; k)[ρ0 ]} dt TrH {Y αj (f ; t)ρ(f ; t)} , (3.71) ∂κj κ=0 0

42

Alberto Barchielli

−

∂2 TrH {Gt1 ∨t2 (f ; k)[ρ0 ]} ∂κ1 ∂κ2 κ=0 Z t1 ∧t2 = dt [hhα1 (t)|hα2 (t)i + TrH {J α1 α2 (f ; t) ρ(f ; t)}] 0 Z t1 Z t∧t2 + dt ds TrH {Y α1 (f ; t)Υ (f ; t, s) ◦ Y α2 (f ; s)[ρ(f ; s)]} 0 0 Z t2 Z t∧t1 + dt ds TrH {Y α2 (f ; t)Υ (f ; t, s) ◦ Y α1 (f ; s)[ρ(f ; s)]} , (3.72) 0

0

where Y α (f ; t)[̺] = Z α (f ; t)̺ + ̺Z α (f ; t)∗ + TrZ {1l ⊗ B α J (f ; t)[̺]} , h ∗ Y α (f ; t) = Z α (f ; t) + Z α (f ; t)∗ + R∗ + C f (t) S ∗ h ∗ i i + C b(t) 1l ⊗ B α R + SC f (t) + C b(t) , h ∗ ∗ i J αβ (f ; t) = R∗ + C f (t) S ∗ + C b(t) 1l ⊗ B α B β R + SC f (t) + C b(t) ,

Proof. From the definition of K (3.52a) and eq. (3.67) we get ∂ Ks f ; k(s) = 1(0,tj ) (s) Y αj (f ; s) , −i ∂κj κ=0 −

h ∂2 Ks f ; k(s) [̺] = 1(0,t1 ∧t2 ) (s) hhα1 (t)|hα2 (t)i̺ ∂κ1 ∂κ2 κ=0 n + TrZ 1l ⊗ B α1 B α2 R + SC f (t) + C b(t) ∗ oi ∗ . ̺ R∗ + C f (t) S ∗ + C b(t)

(3.73a)

(3.73b)

(3.73c)

(3.74)

(3.75)

By (3.51), (3.59) we obtain −i

−

Z t ∂ ∂ = Gt (f ; k) Ks f ; k(s) ◦ Υ (f ; s, 0) ds (−i) ∂κj ∂κj κ=0 κ=0 0 Z t ∂ ds L f (s) ◦ (−i) + , Gs (f ; k) ∂κj κ=0 0

(3.76)

Z t ∂2 ∂2 Gt (f ; k) Ks f ; k(s) ◦ Υ (f ; s, 0) =− ds ∂κ1 ∂κ2 ∂κ1 ∂κ2 κ=0 κ=0 0


Z

t

Ks f ; k(s)

∂ ∂ ◦ (−i) ∂κ ∂κ κ=0 κ=0 1 2 0 Z t ∂ ∂ + ds (−i) Ks f ; k(s) Gs (f ; k) ◦ (−i) ∂κ ∂κ κ=0 κ=0 2 1 0 Z t 2 ∂ . Gs (f ; k) ds L f (s) ◦ − ∂κ ∂κ κ=0 1 2 0

+

ds (−i)

43

Gs (f ; k)

(3.77)

By using (3.63), one can check that the solution of eq. (3.76) is Z t∧tj ∂ −i Gt (f ; k) = ds Υ (f ; t, s) ◦ Y αj (f ; s) ◦ Υ (f ; s, 0) ; ∂κj κ=0 0

(3.78)

Note that the first and second moments turn out to be given by h i Z tj ˜ j , tj ) = Eρ0 X(α dt Ec [TrH {Y αj (f ; t)ρ(f ; t)}] ,

(3.79)

then, this expression can be inserted into (3.77). By applying the expressions (3.76), (3.77) to ρ0 , by taking the trace and by using the definitions (3.73), one gets the final result. ⊓ ⊔

0

i h ˜ 1 , t1 )X(α ˜ 2 , t2 ) Eρ0 X(α Z t1 ∧t2 h i = dt hhα1 (t)|hα2 (t)i + Ec [TrH {J α1 α2 (f ; t) ρ(f ; t)}] 0 Z t1 Z t∧t2 + dt ds Ec [TrH {Y α1 (f ; t)Υ (f ; t, s) ◦ Y α2 (f ; s)[ρ(f ; s)]}] 0 0 Z t2 Z t∧t1 ds Ec [TrH {Y α2 (f ; t)Υ (f ; t, s) ◦ Y α1 (f ; s)[ρ(f ; s)]}] , (3.80) dt + 0

0

3.4 Direct detection The detection scheme The measurement of an observable of the type N (P ; t) can be realized according to the scheme of Figure 1, called direct detection. o/ /o /o /o /o /o o/ / Photocounter o /o /o /o /o /o /o /o System SH /o /o /o /o output O field O O O f (t) O I(t) O O O input field

output field

Fig. 1. Direct detection

44

I(t)

Alberto Barchielli

The system SH is stimulated by some input field, say in a coherent state ψ(f ) or in a mixture of coherent states; then, it emits fluorescence light in various directions. A part of this output field reaches a photoelectron counter which produces some output current I(t) “proportional” to the rate of arrival ˜ (t) the process which counts the photons arof the photons. By denoting by N riving to the photocounter and by using Stieltjes integrals, the output current can be written as Z t ˜ (s) , I(t) = F (t − s) dN (3.81) 0

F (t)

where F is a response function which characterizes the apparatus. A typical choice is γ˜ F (t) = k exp − t ; (3.82) 2

k > 0 and γ˜ > 0 are constants which depend on the apparatus. If we do not consider the time–of–flight from the system SH to the detector and we denote by P the projection which gives the part of the field reaching the counter, we can say that the direct detection scheme described here realizes a continual measurement of the observables N (P ; t), t ≥ 0. There is no conceptual difficulty in considering more counters together, but we prefer to continue to study the case of a single detector. The general results given in the previous sections apply to the present case by taking

L˜ f (t) Y(f ; t) Y (f ; t)

˜ ˜ ˜ (t), • d = 1, X(α, t) ≡ X(t) = N (P ; t), X(α, t) ≡ X(t) =N α α • h (t) = 0, c(t) = 0, b(t) = 0, which gives Z (f ;t) = 0, r(k; t) = 0, • B α = P , with P 2 = P ∗ = P , which gives S k(t) − 1l = eik(t) − 1 P . Then, we obtain

Kt (f ; k)[̺] = L˜ f (t) [̺] + eik(t) Y(f ; t)[̺] ,

(3.83a)

˜ )[̺] = − 1 R∗ R + R∗ SC(f ) + iH ̺ − ̺ 1 R∗ R + C(f )∗ S ∗ R − iH L(f 2 2 ∗ + TrZ 1l ⊗ (1l − P ) R + SC(f ) ̺ R + C(f )∗ S ∗ − kf k2 ̺ , (3.83b) n Y α (f ; t)[̺] ≡ Y(f ; t)[̺] = TrZ 1l ⊗ P R + (S − 1l) C f (t) + C P f (t) h ∗ ∗ io ̺ R∗ + C f (t) (S ∗ − 1l) + C P f (t) , (3.83c)

h ∗ i ∗ Y α (f ; t) ≡ Y (f ; t) = R∗ + C f (t) S ∗ − 1l + C P f (t) h i 1l ⊗ P R + S − 1l C f (t) + C P f (t) ,

(3.83d)


J αβ (f ; t) ≡ J(f ; t) = Y (f ; t)

45

(3.83e)

Usually the detector is placed in a position in which it is not reached by the direct light of the stimulating laser; in mathematical terms we have P f (t) = 0 ,

∀t .

(3.84)

Equations (3.83c) and (3.83d) are written in a way that puts in evidence the terms that disappear when condition (3.84) holds. Moments ˜ , the first two moments (3.79), (3.80) In the case of the counting process N become i Z t h ˜ (t) = dτ Ec [TrH {Y (f ; τ )ρ(f ; τ )}] , (3.85) Eρ0 N 0

i i h h ˜ (t)N ˜ (s) = Eρ N ˜ (t ∧ s) Eρ0 N 0 Z t∨s Z t∧s + dτ1 dτ2 Ec [TrH {Y (f ; τ1 )Υ (f ; τ1 , τ2 ) ◦ Y(f ; τ2 )[ρ(f ; τ2 )]}] t∧s 0 Z t∧s Z τ1 +2 dτ1 dτ2 Ec [TrH {Y (f ; τ1 )Υ (f ; τ1 , τ2 ) ◦ Y(f ; τ2 )[ρ(f ; τ2 )]}] . 0

0

(3.86)

Then, for the output current (3.81) we get Eρ0 [I(t)] = k

Z

t 0

dτ e−˜γ (t−τ )/2 Ec [TrH {Y (f ; τ )ρ(f ; τ )}] ,

Z t∧s t+s dτ e−˜γ ( 2 −τ ) Ec [TrH {Y (f ; τ )ρ(f ; τ )}] Eρ0 [I(t)I(s)] = k2 0 Z t∨s Z τ1 Z t∧s Z t∧s 2 dτ2 e−˜γ (t+s−τ1 −τ2 )/2 dτ2 + 2 dτ1 dτ1 +k t∧s

0

0

(3.87)

0

× Ec [TrH {Y (f ; τ1 )Υ (f ; τ1 , τ2 ) ◦ Y(f ; τ2 )[ρ(f ; τ2 )]}] .

(3.88)

In photocounting problems it is usual to measure the deviations from the variance of the Poisson process by means of the Mandel Q-parameter [19], defined by i i h h ˜ (t) − N ˜ (s) − Eρ N ˜ (t) − N ˜ (s) Var N 0 h i Q(t, s) = , t > s. (3.89) ˜ (t) − N ˜ (s) Eρ0 N

Q(t, s)

46

Alberto Barchielli

For a Poisson process one has Q(t, s) = 0; the terms sub- and super–Poissonian statistics are used for the cases Q(t, s) < 0 and Q(t, s) > 0, respectively. The sub–Poissonian case Q(t, s) < 0 is considered as an index of “nonclassical” light. In our case we obtain the expressions i Z t h ˜ ˜ dτ Ec [TrH {Y (f ; τ )ρ(f ; τ )}] , (3.90) Eρ0 N (t) − N (s) = s

Eρ0

+2

i 2 h ˜ (t) − N ˜ (s) ˜ (t) − N ˜ (s) = Eρ0 N N Z

dτ1

×

s

t

τ1

s

s

Q(t, s) = 2 Z

Z

t

Z Z

t

s τ1

dτ2 Ec [TrH {Y (f ; τ1 )Υ (f ; τ1 , τ2 ) ◦ Y(f ; τ2 )[ρ(f ; τ2 )]}] , (3.91)

−1 dτ E [TrH {Y (f ; τ )ρ(f ; τ )}] c

n dτ2 Ec [TrH {Y (f ; τ1 )Υ (f ; τ1 , τ2 ) ◦ Y(f ; τ2 )[ρ(f ; τ2 )]}] s o c − E [TrH {Y (f ; τ1 )ρ(f ; τ1 )}] Ec [TrH {Y (f ; τ2 )ρ(f ; τ2 )}] . (3.92)

dτ1

In reality the interesting case, for which the Mandel Q-parameter was introduced, is that of a counting process with stationary increments; so, the most useful parameter is Q := lim Q(t, 0) , (3.93) t→+∞

if this limit exists. Probabilities for counts Let us note that the operator L f (t) = Kt (f ; 0) = L˜ f (t) + Y(f ; t)

Υ˜ (f ; t, s)

(3.94)

generates the dynamics Υ (f ; t, s) through eq. (3.63). In constructing the probabilities it is useful to introduce similar maps generated by L˜ f (t) : Υ˜ (f ; t, s)[̺] − ̺ =

Z

s

t

L˜ f (τ ) ◦ Υ˜ (f ; τ, s)[̺] dτ .

(3.95)

Such maps turn out to be positive (even completely positive), but not trace preserving: n o ̺ ≥ 0 ⇒ Υ˜ (f ; t, s)[̺] ≥ 0 , TrH Υ˜ (f ; t, s)[̺] ≤ TrH {̺} . (3.96)


47

By using Υ˜ (f ; t, s) and Y(f ; t), we define the completely positive maps on T (H) Ist (f ; 0) =Υ˜ (f ; t, s) , Z tm Z Z t dtm−1 · · · Ist (f ; m) = dtm s

s

t2

dt1 Υ˜ (f ; t, tm )

s

(3.97)

◦ Y(f ; tm ) ◦ Υ˜ (f ; tm , tm−1 ) ◦ Y(f ; tm−1 ) ◦ · · · ◦ Υ˜ (f ; t2 , t1 ) ◦ Y(f ; t1 ) ◦ Υ˜ (f ; t1 , s) .

Then, one can check that the solution of eq. (3.56) with generator (3.83a) can be written as ∞ X Gst (f ; κ) = eimκ Ist (f ; m) ; (3.98) m=0

in the physical literature solutions of evolution equations written as expansions with respect to a part of the generator are called Dyson series. By the connection (3.65) with the instruments, we immediately identify {Ist (f ; m), m = 0, 1, . . .} as the instrument giving the counts in the time interval (s, t]. Instruments for counts of such a type were introduced by Davies [43, 44]. By combining eqs. (3.66) and (3.97), we can say that the quantities oi h n Pt (0|ρ0 ) = Ec TrH Υ˜ (f ; t, 0)[ρ0 ] , (3.99a) h n pt (tm , tm−1 , . . . , t1 |ρ0 ) = Ec TrH Υ˜ (f ; t, tm ) ◦ Y(f ; tm ) ◦ Υ˜ (f ; tm , tm−1 ) oi ◦ Y(f ; tm−1 ) ◦ · · · ◦ Υ˜ (f ; t2 , t1 ) ◦ Y(f ; t1 ) ◦ Υ˜ (f ; t1 , 0)[ρ0 ] (3.99b)

determine all the probabilities for counts. The quantity (3.99a) is the probability of having no count in the time interval (0, t], when the initial state of the system is ρ0 , and the probability of exactly m counts in the interval (0, t] is Z t Z tm Z t2 Pt (m|ρ0 ) = dtm dtm−1 · · · dt1 pt tm , tm−1 , . . . , t1 |ρ0 . (3.100) 0

0

0

The quantity (3.99b) is the probability density of a count at time t1 , a count at time t2 ,. . . , and no other count in the interval (0, t]; these quantities are called exclusive probability densities and from them also more complicated probabilities can be obtained. Indeed, the characteristic functional (3.45), from which all probabilities can be computed, turns out to be writable as Φt (k) = Pt (0|ρ0 ) +

∞ Z X

m=1

···

t

dtm

0

Z

0

t2

Z

tm

dtm−1

0

X m k(tn ) pt (tm , . . . , t1 |ρ0 ). dt1 exp i n=1

(3.101)

48

Alberto Barchielli

3.5 Optical heterodyne detection Ordinary heterodyne detection Also a different kind of measurement, the so called heterodyne detection [87, 90], can be described by using QSC. By inserting a beam splitter (a half transparent mirror) before the photoelectron counter, the output field from SH is made to beat with an intense laser field (local oscillator); only then, the intensity of the compound beam is measured by the photoelectron counter. This measurement scheme is illustrated in Figure 2.

output port 2

O

I(t) beam splitter O O output /o /o /o /o /o /o /o / O /o /o /o output /o /o /o /o / Photocounter port 1 field O O OO O local oscillator O O O

O

o /o /o o/ output /o /o /o /o System SH O field O O O f (t) O O O O

O

O O O O

O

f0 (t)

Fig. 2. Heterodyne detection

A0 (t) 0 Γ

Let us introduce a new field A0 (t) which does not interact with SH and which can be used to describe the local oscillator; the initial state of this new field is taken to be a coherent vector ψ(ℓf0 ), where ℓ > 0 is a parameter which we shall send to infinity in order to have a very intense laser field. The Fock space is now 0Γ ⊗ Γ , 0 Γ = Γ L2 (R+ ) ; we can also write 0 Γ ⊗ Γ = Γ L2 (R+ )⊗(C⊕Z) . Let us assume that the basis {zk } in Z is chosen in such a way that the index contains the direction of propagation and that only field 1 reaches the beam splitter. So, at the two input ports of the beam splitter the fields Aout 1 (t) and A0 (t) arrive. Let us call B+ (t) and B− (t) the fields leaving the two output ports; they are given by 1 1 B+ (t) = √ Aout 1 (t) + √ A0 (t) , 2 2 1 1 out B− (t) = √ A0 (t) − √ A 1 (t) . 2 2

(3.102)

In other terms the beam splitter operates the unitary transformation (z0 , z1 ) → (z+ , z− ), z± = √12 (z0 ± z1 ). The phases of all the fields can always be


49

redefined in order to have no additional phase shifts in (3.102). Note that the B-fields satisfy the CCR’s as the input and the output A-fields. A photoelectron counter is placed in such a way that it collects the light coming out from one of the output ports of the beam splitter; let us say port 1. The detector counts the photons carried by field B+ (t); then, the number operator “measured” by the photodetector is ∗ N out P± = |z± ihz± | , + (t) = U (t) N (P+ ; t)U (t) , 1 N (P+ ; t) = [Λ11 (t) + Λ10 (t) + Λ01 (t) + Λ00 (t)] 2

(3.103)

The local oscillator is at disposal of the experimenter and its characteristics are known. So, we can subtract from the output of the R t counter the known signal 21 ℓ2 |f0 (t)|2 ; this generates a phase factor − 2i ℓ2 0 k(s)|f0 (s)|ds in the characteristic operator. Moreover, we rescale the output of the counter by a factor 2/ℓ: this amounts to replace k(s) by 2k(s)/ℓ everywhere in the characteristic functional. At the end, the output current is Z t 2 ˜ 2 dN+ (s) − ℓ |f0 (s)| ds , (3.104) F (t − s) I(t) = ℓ 0 which corresponds to the characteristic operator Z t Z 2i t bℓt (k) = exp −iℓ Φ k(s)|f0 (s)|2 ds + k(s)dN (P+ ; s) . ℓ 0 0

(3.105)

In other terms the characteristic operator has the general form discussed in 2 Theorem d = 1,hα = 0, b = 0, r(k; t) = 0, c(t) = −ℓ |f0 (t)| , 3.1 with S k(t) − 1l = e2ik(t)/ℓ − 1 P+ . By applying Proposition 3.2, we obtain that the reduced characteristic operator Gtℓ (f ; k) satisfies eq. (3.51) with generator

X 1 2iκ/ℓ 2 e −1 S1j fj (t) = L f (t) [̺]−iℓκ |f0 (t)| ̺+ R1 + 2 j X X ∗ ∗ × ̺ R1 + S1j fj (t) ̺ fi (t)S1i + ℓf0 (t) R1 +

Ktℓ (f ; κ)[̺]

i

j

X 2 fi (t)S1i∗ ℓf0 (t) + ℓ2 |f0 (t)| ̺ . (3.106) + ̺ R1∗ + i

Now, we consider a very intense laser field for the local oscillator: ℓ → ∞. In this limit the generator of the reduced characteristic operator becomes 2 Kt (f ; κ)[̺] = L f (t) [̺] − κ |f0 (t)| ̺ X X + iκ f0 (t) R1 + S1j fj (t) ̺ + ̺ R1∗ + fi (t)S1i∗ f0 (t) . (3.107) j

i

50

Alberto Barchielli

By comparing (3.107) with the general expression (3.52), we see that the reduced characteristic operator of heterodyne detection is given by a factor corresponding to the measurement of the compatible observables Z th i f0 (s) dA†1 (s) + f0 (s) dA1 (s) , Q(z1 ⊗ f0 ; t) = t ≥ 0, (3.108) 0

times the expression

Z 2 1 t 2 exp − k(s) |f0 (s)| ds ; 2 0

(3.109)

this term represents a classical Gaussian noise to be added to the quantum noise intrinsic to the measurement of Q(z1 ⊗f0 ; t). As we now see, it is possible to eliminate this extra noise. Balanced heterodyne detection The noise in the output current can be reduced by the measurement scheme called balanced heterodyne detection [88]. Now two identical photoelectron counters are used for detecting the photons in both the fields B+ (t) and B− (t) and the difference I1 (t) − I2 (t) of the two output currents is measured; the scheme of balanced heterodyne detection is given in Figure 3.

O I(t)

output

O O

O O

Photocounter 2 O

O O O O O

field

System SH

f (t)

O

O

output port 2

O

O

O

O

O

O

O

O

O

/− O

beam

I1 (t) qqqsplitter q q O qqq O qqqq output o/ o/ o/ /o /o /o /o /o /o /o /o q/ qO q /o /o /o /o /o /o output /o /o /o /o o/ / Photocounter 1 q O port 1 field q q O qq O qqq q q O qq O local oscillator O O O

O

O

I2 (t)

O

f0 (t)

Fig. 3. Balanced heterodyne detection

We scale again the final output current by a factor ℓ−1 , so that we have Z h i 1 t ˜+ (s) − dN ˜− (s) , I(t) = F (t − s) dN (3.110) ℓ 0


51

which corresponds to a continual measurement of the commuting operators Xℓ (t) =

1 [N (P+ ; t) − N (P− ; t)] . ℓ

(3.111)

Then, the characteristic operator Z t exp i k(s) dXℓ (s) 0

has the structure given in Theorem 3.1 with d = 1, c = 0, hα = 0, rt (k) = 0, B α ≡ B = 1ℓ (P+ − P− ), S k(t) − 1l = eik(t)/ℓ − 1 P+ + e−ik(t)/ℓ − 1 P− .

Now we construct the reduced characteristic operator in three steps: first we eliminate the field A0 , then we take the limit of a very intense local oscillator ℓ → ∞, and finally we eliminate the other fields. By recalling that the field A0 does not interact with the system SH , we can do the first two steps by considering the matrix elements on the coherent vectors:

ℓ bt (k)ψ(g 2 ) := ψ(g 1 ) Φ Γ Z t E D 1 k(s) dXℓ (s) ψ(ℓf0 ) ⊗ ψ(g 2 ) 0 ψ(ℓf0 ) ⊗ ψ(g ) exp i 0

Γ ⊗Γ

. (3.112)

Then, by eq. (3.35) we obtain

Z

h 2 ds ℓ2 |f0 (s)| + g11 (s)g12 (s) 0 i . (3.113) × cos k(s)/ℓ − 1 + iℓ f0 (s)g12 (s) + g11 (s)f0 (s) sin k(s)/ℓ

ℓ b (k)ψ(g 2 ) = hψ(g 1 )|ψ(g 2 )i exp ψ(g 1 ) Φ t

t

In the limit of a strong local oscillator we get

ℓ

bt (k)ψ(g 2 ) = hψ(g 1 )|ψ(g 2 )i bt (k)ψ(g 2 ) = lim ψ(g 1 ) Φ ψ(g 1 ) Φ ℓ→∞ Z t h i 2 1 2 2 1 , × exp ds − k(s) |f0 (s)| + ik(s) f0 (s)g1 (s) + g1 (s)f0 (s) 2 0 (3.114)

which is the characteristic operator of the observables Q(z1 ⊗ f0 ; t) (3.108). With the notations of the Theorem 3.1 we have d = 1, B α = 0, c = 0, b = 0, α h (t) ≡ h(t) = f0 (t)z1 , St (k) = 1l, r(k; t) = ik(t)f0 (t)z1 .

(3.115)

Then, the generator of the reduced characteristic operator becomes

52

Alberto Barchielli

1 2 Kt (f ; κ)[̺] = L f (t) [̺] − κ2 |f0 (t)| ̺ 2 X X + iκ f0 (t) R1 + S1j fj (t) ̺ + ̺ R1∗ + fi (t)S1i∗ f0 (t) , (3.116) j

i

which is the similar to the expression (3.107), a part from a smaller noise due to the factor 1/2 in the second term. Moments By using h(t) = f0 (t)z1 , we can say that the balanced heterodyne scheme realizes a continual measurement of the compatible quantum observables i XZ th hzj |h(s)i dA†j (s) + hh(s)|zj i dAj (s) , t ≥ 0 ; (3.117) Q(h; t) = j

0

˜ t) the associated stochastic process, the output current by denoting by Q(h; of the balanced heterodyne detection scheme is Z t ˜ s) . I(t) = F (t − s) dQ(h; (3.118) 0

The generator of the reduced characteristic operator associated to the process ˜ t) is Q(h; κ2 2 kh(t)k ̺ + iκ [Z(f ; t)̺ + ̺Z(f ; t)∗ ] , (3.119) Kt (f ; κ)[̺] = L f (t) [̺] − 2 ∗ Z(f ; t) = C h(t) R + (S − 1l)C f (t) + hh(t)|f (t)i. (3.120) (3.121)

Typical choices are

γ˜ F (t) = k exp − t , 2 ˆ, h(t) = e−iνt h

γ˜ > 0 ,

ˆ ∈Z, h

ˆ (t)i = 0 , hh|f

ν ∈ R, ∀t .

(3.122) (3.123) (3.124)

These choices give

κ2

ˆ 2 Kt (f ; κ)[̺] = L f (t) [̺] −

h ̺ + iκ [Z(f ; t)̺ + ̺Z(f ; t)∗ ] , 2 ˆ ∗ R + (S − 1l)C f (t) . Z(f ; t) = eiνt C h

(3.125) (3.126)


53

By changing ν, which means to change the measuring apparatus, the whole spectrum of SH can be explored. The condition (3.124) means that the light of the laser stimulating SH does not hit the detector directly. The first and second moments turn out to be given by h i Z t ˜ t) = Eρ0 Q(h; ds Ec [TrH {Y (f ; s)ρ(f ; s)}] , (3.127) 0

i Z t1 ∧t2 2 ˜ ˜ Eρ0 Q(h; t1 )Q(h; t2 ) = kh(t)k dt 0 Z t1 ∨t2 Z t1 ∧t2 ds Ec [TrH {Y (f ; t)Υ (f ; t, s) ◦ Y(f ; s)[ρ(f ; s)]}] dt + h

+2

Z

t1 ∧t2 t1 ∧t2

0

dt

Z

0 t

0

ds Ec [TrH {Y (f ; t)Υ (f ; t, s) ◦ Y(f ; s)[ρ(f ; s)]}] , (3.128)

where Y(f ; t)[̺] = Z(f ; t)̺ + ̺Z(f ; t)∗ ,

(3.129)

Y (f ; t) = Z(f ; t) + Z(f ; t) .

(3.130)

∗

By these equations, the measurement scheme we are discussing here can be interpreted as an imprecise, indirect, continual measurement of the system observables Y (f ; t). Then, for the output current (3.118) we get Eρ0 [I(t)] = k

Z

t 0

dτ e−˜γ (t−τ )/2 Ec [TrH {Y (f ; τ )ρ(f ; τ )}] ,

(3.131)

Z t∧s t+s 2 e−˜γ ( 2 −τ ) kh(τ )k dτ Eρ0 [I(t)I(s)] = k2 0 Z t∨s Z τ1 Z t∧s Z t∧s dτ2 e−˜γ (t+s−τ1 −τ2 )/2 dτ1 dτ2 + 2 dτ1 + k2 t∧s

0

0

0

× Ec [TrH {Y (f ; τ1 )Υ (f ; τ1 , τ2 ) ◦ Y(f ; τ2 )[ρ(f ; τ2 )]}] . (3.132)

3.6 Physical models In the next sections we want to present two concrete applications in quantum optics of the whole theory of quantum continual measurements. An interesting phenomenon is the so called shelving effect: a three-level system with a peculiar configuration of permitted transitions and suitably stimulated by lasers exhibits bright and dark periods in its fluorescence light. This phenomenon can have a nice mathematical treatment by using QSC and the theory of continual measurements (direct detection) [16, 20, 42].

54

Alberto Barchielli

Also the simplest quantum system, a two-level atom, presents interesting features. A noteworthy characteristic of its fluorescence spectrum is a three peaked shape in the case of a very intense stimulating laser (Mollow spectrum). Again QSC and the theory of continual measurements give a way of modelling and studying this system [26–28], giving a unified treatment to known results and allowing modifications of the known results by using an Hudson-Parthasarathy equation with S 6= 1l. It is in this problem that it is used a mixture of coherent states of type S3 as initial state [28].

4 A three–level atom and the shelving effect Today experimental techniques allow to observe the fluorescent light emitted by a single atom or ion; therefore, it is possible to observe effects which are completely masked when many emitters are involved. One of these phenomena, the electron shelving effect, was proposed by Dehmelt as a very sensitive scheme for detecting very weak transitions in single ions [45, 46] and it was observed when single ion spectroscopy became feasible [30, 83, 85]. Consider a three–level atom with states |gi (ground), |bi (blue) and |ri (red); assume that the blue transition b ↔ g is very strong and the red one r ↔ g is very weak, while the transition b ↔ r is prohibited. When both transitions are driven by two suitably tuned lasers, we expect the atom to emit blue fluorescent light. But sometimes, when the atom absorbs a red photon, the electron goes into the red state, which has a long lifetime (a fraction of a second), and the fluorescence light stops until the red state decays. Thus we expect to observe bright and dark periods, randomly distributed. In a pictorial language, we say that during a dark period the electron is ‘shelved’ in the red state. The energy–level scheme we have described, when indeed |gi is the lowest state, is called the V configuration and it is given in Figure 4; the |bi≡|1i

22 Y222 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 22 2

|ri≡|2i

@

|gi≡|0i

Fig. 4. Energy–level scheme for the V configuration.


55

simple arrows represent spontaneous decay and the double arrows represent absorption/stimulated emission. The same considerations apply to the so called Λ configuration when |gi is the highest state (Figure 4). Usually the electron jumps between the |gi and the |bi states emitting blue light, but sometimes the |gi state decays into the |ri one and it stays ‘shelved’ there until it absorbs a red photon. A more realistic model could be obtained by adding a weak |bi ↔ |ri transition [91]. |gi≡|0i

>^ > E >>> >>> >> >> >> >> >> >> >> >> >> >> >> > > |ri≡|2i |bi≡|1i

Fig. 5. Energy–level scheme for the Λ configuration.

However, this discussion is of a semi-classical character and does not takes into account that the atom-field interaction gives rise to quantum coherent superpositions of the atomic states. So, to explain the shelving effect we need a full quantum mechanical treatment. A first good quantum–mechanical explanation of this effect was given in [39]. 4.1 The atom–field dynamics Let us concretize the previous discussion in the choice of the operators H, R, S appearing in the Hudson-Parthasarathy equation (2.38) and of the photon space Z; this choice fixes the atom-field dynamics. We denote by |ji, j = 0, 1, 2 the three states: |gi ≡ |0i, |bi ≡ |1i, |ri ≡ |2i. The level scheme of Figures 4 and 5 implies that the free atomic Hamiltonian is ( 2 X ωj > 0 in the V-configuration, ωj |jihj| , H= (4.1) ωj < 0 in the Λ-configuration. j=1 Moreover, we consider the simplest case: only the absorption/emission process is relevant and there is not direct scattering; so we take S = 1l .

(4.2)

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Alberto Barchielli

According to the discussion on the quasi–monochromatic approximation of Section 2.5 we need a different field for any possible atomic transition, if well separated. We have two well separated transitions, the g ↔ b one and the g ↔ r one; so, we take: Z = Z 1 ⊕ Z 2 . Then, in the rotating wave–approximation, the operator R must give the two possible decays with emission of blue photons (in Z 1 ) and red photons (in Z 2 ); this implies ( |0 ⊗ αj ihj| in the V-configuration, 1 2 j R=R +R , R = (4.3) |j ⊗ αj ih0| in the Λ-configuration, αj ∈ Z j ,

2 γj := αj > 0 .

(4.4)

The two following quantities constructed from R appear in the master equation (2.73): (P 2 in the V-configuration, ∗ j=1 γj |jihj| (4.5) R R= (γ1 + γ2 )|0ih0| in the Λ-configuration, (P2 j=1 γj hj|̺|ji |0ih0| in the V-configuration, ∗ TrZ {R̺R } = (4.6) P2 h0|̺|0i j=1 γj |jihj| in the Λ-configuration, Pure decay

In order to understand better the meaning of the various quantities appearing in the atom–field dynamics, let us start with the case of the atom not stimulated in any way; this correspond to the choice S1 for the state: the initial field is in the Fock vacuum (f ≡ 0). By eqs. (2.68), (2.69) we have ρ(t) = ρ(0; t) and this state satisfies the master equation (2.75). This equation can be easily written in our case and solved; the final result is: 1. In the case of the V-configuration: i, j = 1, 2, hj|ρ(0; t)|ii = exp − 21 (γi + γj ) t + i (ωi − ωj ) t hj|ρ0 |ii, hj|ρ(0; t)|0i = h0|ρ(0; t)|ji = exp − 21 γj t − iωj t hj|ρ0 |0i, h0|ρ(0; t)|0i = 1 −

2 X j=1

e−γj t hj|ρ0 |ji.

(4.7a) (4.7b) (4.7c)

2. In the case of the Λ-configuration: i, j = 1, 2, h0|ρ(0; t)|0i = e−(γ1 +γ2 )t h0|ρ0 |0i, (4.8a) 1 hj|ρ(0; t)|0i = h0|ρ(0; t)|ji = exp − 2 (γ1 + γ2 ) t − iωj t hj|ρ0 |0i, (4.8b) γj 1 − e−(γ1 +γ2 )t h0|ρ0 |0i. hj|ρ(0; t)|ii = ei(ωi −ωj )t hj|ρ0 |ii + δij γ1 + γ2 (4.8c)


57

These equations confirm that ω1 , ω2 are the atomic frequencies and show that γ1 , γ2 are the spontaneous emission rates. The assumption of a weak red transition and a strong blue one gives γ1 ≫ γ2 > 0. Stimulating lasers Let us consider now the case of interest, when we have two lasers stimulating the blue and the red transitions. This is the case of a state of type S2, with the field in a coherent state with f of the form f (t) = 1l[0,T ] (t)

2 X

e−iνj t λj ,

j=1

νj > 0 ,

λj ∈ Z j .

(4.9)

In all the following formulae we take T → +∞. By eqs. (2.68), (2.69), we have ρ(t) ≡ ρ(f ; t); this reduced state satisfies the master equation (2.73), with Liouvillian (2.74) 1 ∗ 1 ∗ L f (t) [̺] = − R R + iH f (t) ̺ − ̺ R R − iH f (t) 2 2

+ TrZ {R̺R∗ } ,

(4.10)

where (cf eq. (2.72c)) ∗ H f (t) = H + iC f (t) R − iR∗ C f (t) 2 n o X j j ωj |jihj| + ieiνj t hλj |αj iσ− − ie−iνj t hαj |λj iσ+ , =

(4.11)

j=1

j σ− := |0ihj| ,

j σ−

:= |jih0| ,

j σ+ := |jih0| ,

j σ+

:= |0ihj| ,

in the V-configuration, in the Λ-configuration.

(4.12)

The explicit time dependence due to the lasers can be removed; in the physical parlance one says that the rotating frame is used. Let us set ( +1 in the V-configuration, (4.13) ǫ := −1 in the Λ-configuration, ∆j := νj − |ωj | , note that one has

Ωj := 2 hαj |λj i ,

βj := ǫ arg −ihαj |λj i ; (4.14)

i ǫiβj e Ωj . (4.15) 2 The ∆j are called the detuning parameters and the Ωj are called Rabi frequencies. To have a strong blue transition, strongly stimulated, and a weak red one we assume: hαj |λj i =

58

Alberto Barchielli

γ1 ≫ γ2 > 0 ,

γ1 ≫ Ω2 ,

Ω1 ≫ Ω 2 > 0 ,

Ω12 ≫ γ1 γ2 .

(4.16)

The approximation of taking two field types, one interacting with the blue transition and one with the red one, is reasonable only if the two lasers are not too much out of resonance; moreover, we want the two transitions to be not overdamped. These two conditions give the further assumptions |∆j | . 2γj ,

2Ωj > γj ,

j = 1, 2.

(4.17)

Then, the new reduced state X 2 2 X ρ˜(t) := exp ǫit νj |jihj| ρ(t) exp −ǫit νj |jihj| j=1

(4.18)

j=1

satisfies the master equation d ρ˜(t) = L∆ [˜ ρ(t)] , dt

(4.19)

where L∆ [̺] := K̺ + ̺K ∗ + TrZ {R̺R∗ } , 2 X Ωj iβj 1 K := − R∗ R − i e |jih0| + e−iβj |0ihj| − ǫ∆j |jihj| . 2 2 j=1

(4.20) (4.21)

4.2 The detection process Now we assume to have a detector able to count photons flying through a solid angle Sd not containing the direction of propagation of the lasers, so that the lasers do not send light directly to the counter and only fluorescence light is detected; the efficiency of the counter can be taken into account by choosing Sd smaller than the geometrical solid angle spanned by the detector. We can formalize this setup by saying that the detector performs a continual measurement of the observable N (P ; t) defined by eq. (3.1) with 1. P λj = 0, j = 1, 2: the laser light does not impinges directly on the detector; 2. P Z j ⊂ Z j , j = 1, 2: the detector does not mix up blue and red photons. Let us set

2 ηj := P αj ;

(4.22)

by their definitions, we have 0 ≤ ηj ≤ γj . We assume that “many” blue photons are detected, so we have 0 ≤ η2 ≤ γ2 ≪ η1 ≤ γ1 .

(4.23)


59

The whole information on the counting probabilities is contained in the characteristic functional (3.45), (3.101) or in the probabilities of no counts (3.99a) and the exclusive probability densities (3.99b). To compute these probabilities we need to particularize to our case the quantities of eqs. (3.83); we get Y(f ; t)[̺] = TrZ {(1l ⊗ P ) R̺R∗ } =: Y[̺] , L˜ f (t) = L f (t) − Y

(4.24b)

L˜∆ := L∆ − Y .

(4.25)

n o ˜ Pt (0|ρ0 ) = TrH eL∆ t [ρ0 ]

(4.26)

(4.24a)

From the expression of the probabilities (3.99a), (3.99b) we see that we can make everywhere the transformation (4.18) without changing such probabilities. This simply means that everywhere we have to make the substitutions L f (t) → L∆ and L˜ f (t) → L˜∆ , where In this way we get

for the probability of no counts up to time t and n ˜ ˜ pt (tm , tm−1 , . . . , t1 |ρ0 ) = TrH eL∆ (t−tm ) ◦ Y ◦ eL∆ (tm −tm−1 )

o ˜ ˜ ◦ Y ◦ · · · ◦ eL∆ (t2 −t1 ) ◦ Y ◦ eL∆ t1 [ρ0 ]

(4.27)

for the exclusive probability densities. More concretely we have

L˜∆ [̺] = K̺ + ̺K ∗ (P2 j=1 (γj − ηj ) hj|̺|ji |0ih0| + P2 h0|̺|0i j=1 (γj − ηj ) |jihj| 2 iX Ωj eiβj |jih0| + e−iβj |0ihj| 2 j=1 (P γj 2 j=1 i∆j − 2 |jihj| + P2 2 |0ih0| −i j=1 ∆j |jihj| − γ1 +γ 2

in the V-configuration, in the Λ-configuration,

(4.28)

K=−

κ(̺) := ρjump

(P 2

Y[̺] = κ(̺)ρjump ,

j=1 ηj hj|̺|ji (η1 + η2 ) h0|̺|0i

( |0ih0| := P2

in the V-configuration, in the Λ-configuration,

ηj j=1 η1 +η2

|jihj|

in the V-configuration, in the Λ-configuration, in the V-configuration, in the Λ-configuration.

(4.29)

(4.30a) (4.30b) (4.30c)

60

Alberto Barchielli

By inserting these expressions into eq. (4.27) and setting ˜ ˜ w0 (t) := κ eL∆ t [ρ0 ] , w(t) := κ eL∆ t [ρjump ] ,

(4.31)

we obtain

pt (tm , . . . , t1 |ρ0 ) = Pt−tm (0|ρjump ) w(tm − tm−1 ) · · · w(t2 − t1 ) w0 (t1 ) . (4.32) Moreover, one can check immediately that d ˜ Pt (0|̺) = −κ eL∆ t [̺] dt

(4.33)

and, then, one has

Pt (0|ρjump ) = 1 −

Z

t

w(s)ds ,

0

Pt (0|ρ0 ) = 1 −

Z

t

w0 (s)ds .

(4.34)

0

These equations say that our detection process is a delayed renewal counting process (an usual renewal counting process when w0 (t) ≡ w(t)); w(t) is the R +∞ interarrival waiting–time density. By construction 0 w(t) dt ≤ 1, but it R +∞ is possible to have 0 w(t) dt < 1: this means that there is a non–zero probability that the detected fluorescence stops. Note that the probability density w(t) of the interarrival times is a continuous function and that w(0) = 0; this says that just after a count the atom cannot emit, but it needs some time to be excited again: this is the so called antibunching effect. From now on we assume that all the fluorescence photons are detected; we can think to have an array of perfect detectors spanning the whole solid angle around the atom, with the exception of a small angle containing the direction of propagation of the lasers. In other terms we give the process which counts all the emitted photons. Mathematically this amounts in taking P = 1l

or

ηj = γj ,

j = 1, 2 .

(4.35)

This simplifies all the computations, because we get L˜∆ [̺] = K̺ + ̺K ∗ , Pt (0|ρjump ) = 1 − Pt (0|ρ0 ) = 1 −

Z

t

0

Z

0

t

˜

∗

eL∆ t [̺] = eKt ̺eK t ,

(4.36)

o n ∗ w(s)ds = Tr eKt ρjump eK t ,

(4.37a)

n o ∗ w0 (s)ds = Tr eKt ρ0 eK t .

(4.37b)

As far as many blue photons are detected and (4.23) holds, the assumption (4.35) should not alter the statistical properties of the counting process in an essential way.


61

4.3 Bright and dark periods: the V-configuration As we have seen, all the probabilities depend on the waiting time densities w0 (t) and w(t). By particularizing eqs. (4.29)–(4.31) to the V case, we get K=

2 X j=1

i∆j −

γj i |jihj| − Ωj eiβj |jih0| + e−iβj |0ihj| , 2 2

κ(̺) =

2 X j=1

w0 (t) =

γj hj|̺|ji, 2 X j=1

w(t) =

2 X j=1

ρjump = |0ih0|,

E D ∗ γj j eKt ρ0 eK t j ,

(4.38)

(4.39)

(4.40a)

2 γj j eKt 0 .

(4.40b)

To obtain the expression of the interarrival time density w(t), we have to compute eKt |0i. By setting a0 (t) := h0|eKt |0i ,

aj (t) := e−iβj hj|eKt |0i ,

we can write w(t) =

2 X j=1

Pt (0|ρjump ) = 1 −

Z

j = 1, 2 ,

2

γj |aj (t)| ,

t

w(s)ds =

0

2 X j=0

(4.41)

(4.42)

2

|aj (t)| .

(4.43)

By taking the time derivative, we get the system of linear differential equations   1 d (4.44) a(t) = G a(t) , a(0) = 0 , dt 0

where

 0 −iΩ1 /2 −iΩ2 /2 0 , G := −iΩ1 /2 −ξ1 /2 −iΩ2 /2 0 −ξ2 /2 

ξj := γj − 2i∆j ,

j = 1, 2 .

(4.45) (4.46)

The solution of the system (4.44) can be obtained by Laplace transform; we get

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Alberto Barchielli

(z0 + ξ1 /2)(z0 + ξ2 /2) z0 t e (z0 − z1 )(z0 − z2 ) (z1 + ξ1 /2)(z1 + ξ2 /2) z1 t (z2 + ξ1 /2)(z2 + ξ2 /2) z2 t e + e , (4.47a) + (z1 − z0 )(z1 − z2 ) (z2 − z0 )(z2 − z1 )

a0 (t) =

a1 (t) = −

iΩ1 2

iΩ2 a2 (t) = − 2

z0 + ξ2 /2 ez0 t (z0 − z1 )(z0 − z2 ) z2 + ξ2 /2 z1 + ξ2 /2 ez1 t + ez2 t , + (z1 − z0 )(z1 − z2 ) (z2 − z0 )(z2 − z1 )

(4.47b)

z0 + ξ1 /2 ez0 t (z0 − z1 )(z0 − z2 ) z2 + ξ1 /2 z1 + ξ1 /2 z1 t z2 t e + e , + (z1 − z0 )(z1 − z2 ) (z2 − z0 )(z2 − z1 )

(4.47c)

where z0 , z1 , z2 are the roots (which we have assumed to be all distinct) of the characteristic polynomial of the matrix G: z3 +

1 1 1 (ξ1 + ξ2 ) z 2 + Ω12 + Ω22 + ξ1 ξ2 z + Ω12 ξ2 + Ω22 ξ1 = 0 . (4.48) 2 4 8

By exploiting the assumptions (4.16) and (4.17), we can find an approximate expression for the three roots of this equation: Ω 2 ξ1 ξ2 − 2 2 2 2Ω1 2Ω12 2Ω12 + Ω22 − ξ12 Ω12 + Ω22 Ω22 ξ1 p 1− 2 +i z1 ≃ − , 4 Ω1 Ω12 4Ω12 − ξ12 2Ω12 2Ω12 + Ω22 − ξ12 Ω12 + Ω22 Ω22 ξ1 p . 1− 2 −i z2 ≃ − 4 Ω1 Ω12 4Ω12 − ξ12 Note that one has Re 4Ω12 − ξ12 = 4Ω12 − γ12 + 4∆12 > 0 and z0 ≃ −

0 > Re z0 ≫ max {Re z1 , Re z2 } .

(4.49a) (4.49b) (4.49c)

(4.50)

We can see that, for the values of interest of the parameters, the three roots are indeed distinct and with strictly negative real parts. Using this in eq. (4.43), we get Z +∞ lim Pt (0|ρjump ) = 0 , w(s)ds = 1 . (4.51) t→+∞

0

Similar considerations apply to Pt (0|ρ0 ) and w0 (t). This implies that the fluorescence light never stops definitively.


63

By (4.42), (4.47), many decay times appear in the expression of w(t); by −1 (4.50), the longest one is (−2 Re z ) , while the others, (− Re z − Re zj )−1 , 0 0 −1 (− Re zi − Re zj ) , i, j = 1, 2 are much shorter; so, w(t) has a small long living tail. Indeed, by (4.42) and (4.47), we can write w(t) = wshort (t) + Π2 |Re z0 | e2 Re z0 t ,

(4.52)

where in wshort (t) we have grouped all the terms with a short decay time and 2

Π :=

γ1 Ω12 |z0 + ξ2 /2| + γ2 Ω22 |z0 + ξ1 /2| 2

2

8 |z0 − z1 | |z0 − z2 | |Re z0 |

2

;

(4.53)

moreover, the conditions (4.16) give Π ≪ 1. Therefore, the interarrival times are usually very short, but sometimes (with probability Π after each emission) the interarrival time is long, with a mean of the order of (−2 Re z0 )−1 and the fluorescence light stops for a detectable interval of time. In conclusion we have a sequence of bright and dark periods of random length controlled by w(t); in particular, the mean length of the dark periods is approximately (−2 Re z0 )−1 ≃

Ω12 . Ω12 γ2 + Ω22 γ1

(4.54)

4.4 Bright and dark periods: the Λ-configuration In the Λ case we can proceed in the same way. From eqs. (4.29)–(4.31) we get K=−

γ1 + γ2 |0ih0| 2 −i

2 X Ωj iβj e |jih0| + e−iβj |0ihj| , ∆j |jihj| + 2 j=1

κ(̺) = (γ1 + γ2 ) h0|̺|0i,

ρjump =

2 X j=1

γj |jihj|, γ1 + γ2

E D ∗ w0 (t) = (γ1 + γ2 ) 0 eKt ρ0 eK t 0 , w(t) =

2 X j=1

2 D ∗ E 2 2 X γj 0 eKt j = γj j eK t 0 .

(4.55)

(4.56)

(4.57a) (4.57b)

j=1

The case of equal detunings: the dark state The case ∆1 = ∆2 =: ∆ is very peculiar, because in this situation K has an eigenvector with a purely imaginary eigenvalue. One can check that

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Alberto Barchielli

K ∗ |ϕ2 i = i∆|ϕ2 i,

K|ϕ2 i = −i∆|ϕ2 i, where |ϕ2 i := p

We have also

1 Ω12

+

Ω22

Ω2 eiβ1 |1i − Ω1 eiβ2 |2i . L˜∆ [|ϕ2 ihϕ2 |] = 0 ;

L∆ [|ϕ2 ihϕ2 |] = 0 ,

(4.58)

(4.59)

(4.60)

therefore, the state ϕ2 is perfectly stationary and in this state the atom cannot emit: ϕ2 is a dark state. It is useful to construct an orthonormal set {ϕ0 , ϕ1 , ϕ2 } containing the stationary state ϕ2 ; we choose |ϕ0 i := |0i,

|ϕ1 i := p

1 Ω12

+

Ω22

Ω1 eiβ1 |1i + Ω2 eiβ2 |2i .

(4.61)

By using this c.o.n.s. and by setting x(t) := hϕ0 |eKt |ϕ1 i ,

y(t) := hϕ1 |eKt |ϕ1 i ,

(4.62)

we obtain from eqs. (4.57b), (4.37a) γ1 Ω12 + γ2 Ω22 2 |x(t)| , Ω12 + Ω22 w(t) γ1 Ω12 + γ2 Ω22 2 Pt (0|ρjump ) = Π + + |y(t)| , γ1 + γ2 (γ1 + γ2 ) (Ω12 + Ω22 ) w(t) =

where Π :=

2 X j=1

γj γ1 Ω22 + γ2 Ω12 2 |hj|ϕ2 i| = . γ1 + γ2 (γ1 + γ2 ) (Ω12 + Ω22 )

(4.63) (4.64)

(4.65)

By time differentiation we get the linear system q γ1 + γ2 i dx(t) =− x(t) − Ω12 + Ω22 y(t) , dt 2 2 q dy(t) i =− Ω12 + Ω22 x(t) − i∆y(t) ; dt 2

(4.66)

the initial conditions are x(0) = 0, y(0) = 1. The roots of its characteristic equation are   s 2 1 γ1 + γ2 γ + γ 1 2 z± = −i∆ − (4.67) ± i Ω12 + Ω22 − − i∆  2 2 2 and its solution is


p i Ω12 + Ω22 z− t x(t) = e − ez+ t , 2 (z+ − z− ) 1 y(t) = (z+ + i∆) ez− t − (z− + i∆) ez+ t . z+ − z−

65

(4.68a) (4.68b)

From these results we have limt→+∞ Pt (0|ρjump ) = Π and so, after each emission, there is a probability Π that the fluorescence stops and a probability 1 − Π of a new emission. By (4.37a) we have also Z

+∞

w(t)dt = 1 − Π =

0

γ1 Ω12 + γ2 Ω22 . (γ1 + γ2 ) (Ω12 + Ω22 )

(4.69)

Let us consider for simplicity the case ρ0 = ρjump , which gives w0 (t) = w(t). If Ntot denotes the total number of emissions before the light stops, by the renewal structure we have P [Ntot = n] = (1−Π)n Π, n = 0, 1, 2, . . .. Moreover, the inter-emission times Ti are independent and identically distributed random variables with probability density 2 (γ1 + γ2 ) Ω12 + Ω22 z− t w(t) e − ez+ t . (4.70) = 2 1−Π 4 |z+ − z− | Then, T := is

PNtot i=1

Ti is the duration of the initial bright period and its mean

1−Π E[T ] = E[Ntot ] E[T1 ] = Π

Z

+∞

0

w(t) 1 t dt = 1−Π Π

Z

+∞

tw(t)dt . (4.71)

0

The case of different detunings In the generic case ∆1 6= ∆2 we can write the expression (4.42) for w(t) by setting ∗

a0 (t) := h0|eK t |0i ,

∗

aj (t) := e−iβj hj|eK t |0i ,

j = 1, 2 .

(4.72)

Then, we have to solve the system of linear differential equations (4.44) where now   −(γ1 + γ2 )/2 iΩ1 /2 iΩ2 /2 iΩ1 /2 i∆1 0 . G :=  (4.73) iΩ2 /2 0 i∆2

One can show that, when ∆1 6= ∆2 , all the eigenvalues of G have not vanishing negative real parts and that, under conditions (4.16), one of the decay times in w(t) is very long and the other ones very short; then, the discussion on bright and dark periods goes on in a similar way as in the V-system case [42].

66

Alberto Barchielli

5 A two–level atom and the spectrum of the fluorescence light The simplest, non trivial matter-field system is a two-level atom stimulated by a laser. The fluorescence spectrum of such a system, in the case of a perfectly monochromatic laser, was obtained by Mollow [81]; then, the case of a laser with a Lorentzian spectrum was developed by Kimble and Mandel [68,75]. The Hudson-Parthasarathy equation (2.38) gives not only the way of giving an unified treatment of such a system and of connecting it to the photon-detection theory, but it allows also to explore possible corrections to the dynamics and to obtain modifications of the Mollow spectrum. In the usual treatments, the scattering of the light by the atom is described by the absorption/emission process due to the electric dipole interaction involving only two states of the atom. But even if the absorption/emission process would be forbidden and the atom would be frozen in the up or down level, some scattering of light would remain, due to the response of the atom as a whole; we can call it “direct scattering”. For instance, in a perturbative development in Feynman graphs, scattering processes would be generated also by virtual transitions starting and ending in one of the two states left in the description, but involving as intermediate states the other ones, which have been eliminated in the final description. The use of Hudson-Parthasarathy equation allows, even when the atom is approximated by a two-level system, for the introduction of both the “absorption/emission channel”, through the terms containing the R operator, and the “direct scattering channel”, through the terms containing the S operator. This modified model, in the case of a perfectly monochromatic laser, has been presented in [26] and developed in [27]; the results for a non monochromatic laser have been obtained in [28]. Being a relatively simple situation, we want to take the opportunity of showing also how the polarization of light can be introduced in QSC; to be consistent, also the electromagnetic selection rules have to be taken into account and the up and down levels have to be degenerate. 5.1 The dynamical model The two-level atom The experiments [41, 49, 59, 60, 86] involve or the hyperfine component of the D2 line of sodium with up level 2P3/2 , F = 3 and down level 2S1/2 , F = 2, or the levels 6s6p 1 P1 , F = 1, and 6s2 S0 , F = 0 of 138 Ba; F is the total angular momentum. In order to describe an atom with two degenerate levels as in the experimental situation, we take H = C2F− +1 ⊕ C2F+ +1 ,

F+ = F − + 1 ,

where F− is integer or semi-integer. We denote by

(5.1)


|F± , M i ,

67

M = −F± , . . . , F± ,

the angular momentum basis in H; the parities of the states of the two levels must be opposed, let us say they are ǫ± , with ǫ+ ǫ− = −1. Let us denote by P± =

F± X

M =−F±

|F± , M ihF± , M |

(5.2)

the two projection on the up or down states. Then, the energy H of the free atom, contained in the quantity K (2.36e), must be given by H=

1 ω0 (P+ − P− ) , 2

ω0 > 0 ,

(5.3)

where the atomic frequency ω0 must already include the Lamb shift. The photon space In the approximations we are considering [18,89], the fields behave as monodimensional waves, so that a change of position is equivalent to a change of time and viceversa. Then, the space Z has to contain only the degrees of freedom linked to the direction of propagation and to the polarization. To describe a spin-1 0-mass particle we use the conventions of Messiah [79] pp. 550, 1032– 1037. The space Z is spanned by the c.o.n.s. ~̟ , |j, m; ̟i ≡ Θ jm

j = 1, 2, . . . ,

m = −j, −j + 1, . . . , j,

̟ = ±1 ;

jm is the total angular momentum, ̟ = +1 denotes the electrical multipoles, ̟ = −1 denotes the magnetic multipoles and (−1)j ̟ is the parity. By using the spherical harmonics Ylm (θ, φ) and the orbital angular momentum operator ~ℓ, one has ~ℓ Yjm , ~ −1 = p 1 ~ −1 , ~ +1 = i~ Θ (5.4) p×Θ Θ jm jm jm j(j + 1) where p~ is the direction versor given by p1 = sin θ cos φ ,

p2 = sin θ sin φ ,

p3 = cos θ .

(5.5)

The interaction Let us consider now the terms with the creation and annihilation operators in the dynamical equation (2.38); they must describe the absorption/emission process. By asking spherical symmetry, parity conservation and only electrical dipole contribution in R, we must have

68

Alberto Barchielli F+ X

R=α

M =−F+

|ǫ+ ; F− ; 1; F+ , M ihF+ , M |,

α ∈ C,

α 6= 0 ,

(5.6)

|ǫ+ ; F− ; 1; F+ , M i =

F− X

1 X

m1 =−F− m2 =−1

|F− , m1 i ⊗ |1, m2 ; +1ihF− , m1 ; 1, m2 |F+ , M i,

(5.7)

where by hF− , m1 ; 1, m2 |F+ , M i we denote the Clebsch-Gordan coefficients ([79] pp. 560–563). The interaction terms containing R are responsible for the spontaneous decay of the atom and, as we shall see, |α|2 turns out to be the natural line width. By eqs. (2.36) and (5.2) we have Rk = α

1 X

hzk |1, m; +1iQm ,

(5.8)

m=−1

Qm =

F− X

m1 =−F−

|F− , m1 ihF− , m1 ; 1, m|F+ , m1 + mihF+ , m1 + m|, R∗ R = |α|2 P+ ,

R k = P− R k P+ , X

2

Rk ρRk∗ = |α|

1 X

(5.9) (5.10)

Qm ρQ∗m ,

(5.11)

2 i |α| K − ω0 1l = − iω0 + P+ . 2 2

(5.12)

k

m=−1

The interaction term containing the Λ-process must give the residual scattering when the atom is frozen in the up or down level, so we take the unitary operator S of the form S = P+ ⊗ S + + P− ⊗ S − ,

S ± ∈ U(Z) .

(5.13)

Then, by spherical symmetry and parity conservation we must have S± =

j ∞ X X

X

j=1 m=−j ̟=±1

|j, m; ̟i exp{2iδ± (j; ̟)}hj, m; ̟|,

(5.14)

0 ≤ δ± (j; ̟) < 2π. The unitary operators S + and S − represent the scattering operators for a photon impinging on the atom frozen in the up or in the down state; the δ’s are the phase shifts for these scattering processes. Quantities like ω0 , α, δ± (j; ̟) are phenomenological parameters, or, better, they have to be computed from some more fundamental theory, such as some approximation to quantum electrodynamics.


69

Summarizing, the possible processes are: absorption, through the term containing R∗ , emission, through R, photon scattering with the atom in the up level, through P+ ⊗ S + , photon scattering with the atom in the down level, through P− ⊗ S − . An illustration of these processes is given in Figure 6. 'g 'g w7 7 P+ ⊗S + g' g' w7 7w 7 w g' g' w7 g' 'g w7 w7 7w 'O g' g' 'g 'g R∗ 'g g' g' g' 'g 'g g' g' g' g' g' g' R '7 \ ' ' g w 7 g' g' 7w w 7 g' 'g 7w 'g g' 7w 7w − g' ' 7w 7w P− ⊗S Fig. 6. The dynamical processes: absorption, emission and direct photon scattering

The balance equation for the number of photons Let us introduce the observables, which we have already considered in Section 3.1 eqs. (3.1) and (3.12), “total number of photons in the time interval [0, t]” before and after the interaction with the atom, X in Ntot (t) := N (1l; t) = Λkk (t) , (5.15) k out out ∗ in Ntot (t) := N (1l; t) = U (t) Ntot (t)U (t) . By eq. (3.12) we obtain in the present model the balance equation in Ntot (t) +

1 1 out (P+ − P− ) = Ntot (t) + U (t)∗ (P+ − P− )U (t) . 2 2

(5.16)

Such an equation has a very important meaning: it says that the number of photons entering the system up to time t plus the photons stored in the atom at time 0 is equal to the number of photons leaving the system up to time t plus the photons stored in the atom at time t. The phase diffusion model for the laser In [68,75] a laser model is considered which, translated in our setup, amounts in taking as the state of the laser a mixture of coherent vectors of the type S3, (2.64). Therefore, the initial state is taken to be ρ0 ∈ S(H) , (5.17) s = ρ0 ⊗ Ec η(f ) , f (t) = e−i(ωt+

√ B W (t))

1(0,T ) (t) λ ,

λ∈Z,

ω > 0,

B ≥ 0,

(5.18)

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Alberto Barchielli

W (t) is a real standard Wiener process canonically realized in the Wiener probability space (Ω c , F c , P c ) (S3’); T is a large time and T → +∞ in the final results is always understood. This is the simplest model for a laser which is not perfectly monochromatic nor perfectly coherent. Only the phase fluctuates, not the intensity; moreover, the laser spectrum has a Lorentzian shape with bandwidth B: Z ~ω +∞ iντ c B/(2π) ; (5.19) e E [hf (t)|f (t + τ )i] dτ = ~ωkλk2 2π −∞ (ν − ω)2 + B 2 /4 ~ωkλk2 is the power of the laser. The whole model is meaningful only for ω not too “far” from ω0 . In some experiments the laser light is taken to be circularly polarized, because in this way, in the long run, only two states are involved in the dynamics ([60] p. 206) and the usual theory has been developed for systems with only two states [68,81]. Let the incoming light have right circular polarization and propagate along the z axis; then, the electromagnetic selection rules imply that the atomic transitions which survive are: spontaneous emission with ∆M = 0, ±1, stimulated emission with ∆M = −1, absorption with ∆M = 1; the situation is summarized in the Figure 7. M =−2

M =−1

M =1

M =0

88 88 C 88 88 88 88 88 88 88 888 88 88 88 8 M =−1

M =2

88 C C 88 88 88 8 8 888 8

M =0

M =1

Fig. 7. Allowed atomic transitions (case F+ = 2, F− = 1)

In order to describe a well collimated laser beam propagating along the direction z (θ = 0) and with right circular polarization, we take λ = αΩ eiδ λ+ , ~λ+ (θ, φ) =

∆θ

p

Ω > 0,

1[0,∆θ] (θ) 3π(1 − cos ∆θ)

δ ∈ [0, 2π) 1 ~ ~ −√ i + ij ; 2

(5.20) (5.21)

in all the physical quantities the limit ∆θ ↓ 0 will be taken. Note that the power of the laser ~ωkλk2 = 32 ~ω|α|2 Ω 2 /(∆θ)2 diverges for ∆θ ↓ 0, because we need a not vanishing atom-field interaction in the limit. In the following we shall need the relation r 2j + 1 hj, m; ̟|λ+ i = −̟ δm,1 . (5.22) 12


71

5.2 The master equation and the equilibrium state Let us start by considering the (not averaged) reduced statistical operator ρ(f ; t) = TrΓ {U (t) (ρ0 ⊗ η(f )) U (t)∗ } (2.69), which satisfies the master equation (2.71) or (2.73); the time-dependent Liouvillian (2.72) turns out to be L f (t) [̺] = −i H f (t) , ̺ ∞ ∗ i ∗ i h 1 X h + , + Rk f (t) , ̺Rk f (t) Rk f (t) ̺, Rk f (t) 2

(5.23)

k=1

1 X X hzk |(S ǫ − 1l)f (t)iPǫ , hzk |1, m; +1iQm + Rk f (t) = α m=−1

(5.24)

ǫ=±

1

H f (t) = ω0 − Im f (t) S + − S − f (t) (P+ − P− ) 2 1 i X h α 1 + e−2iδ− (1;+1) hf (t)|1, m; +1iQm + 2 m=−1 i − α 1 + e2iδ− (1;+1) h1, m; +1|f (t)iQ∗m .

(5.25)

We see that the transitions between the two levels are due to the presence of the operators Qm (5.9); more precisely, the transitions from the up to the down level are due to the presence of Qm on the left of ̺ and of Q∗m on the right (emission), while Q∗m on the left and Qm on the right give the transitions from the down to the up level (absorption). The fact that the operators Rk f (t) contain a sum of terms proportional to Qm and of terms proportional to P± is an indication of an interference effect between absorption/emission and direct scattering. Note that, when the field is in the Fock vacuum, only spontaneous emission is present; indeed, for f (t) = 0, the Liouvillian (5.23) reduces to L(0)[̺] = −iω0 [P+ , ̺] −

1 X 1 2 Qm ̺Q∗m , |α| (P+ ̺ + ̺P+ ) + |α|2 2 m=−1

(5.26)

which describes the atomic decay according to the usual selection rules for electric dipole and fixes the meaning of |α|2 as spontaneous decay rate or natural line width. By inserting the expression (5.18), (5.20), (5.21) of f (t) into eqs. (5.24), (5.25) we get

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Alberto Barchielli

√ Rk f (t) = e−i ωt+ B W (t) UW (t)∗ D(zk )UW (t) , ˜ W (t) + 1 ω (P+ − P− ) , H f (t) = UW (t)∗ HU 2 h i √ i ωt + B W (t) (P+ − P− ) , UW (t) := exp 2 D(h) := α

1 X

hh|1, m; +1iQm + αΩeiδ

m=−1

X

ǫ=±

hh| (S ǫ − 1l) λ+ iPǫ ,

h∈Z, h i 1 2 ˜ = ω0 − ω − |α| Ω 2 Imhλ+ | S + − S − λ+ i (P+ − P− ) H 2 h i i 2 + |α| Ω eiδ 1 + e2iδ− (1;+1) Q∗1 − e−iδ 1 + e−2iδ− (1;+1) Q1 . 4

(5.27) (5.28) (5.29)

(5.30)

(5.31)

This result suggests to consider the random atomic state in the “rotating frame” ρ˜(f ; t) := UW (t)ρ(f ; t)UW (t)∗ . (5.32) By classical stochastic calculus we obtain the stochastic equations of Ito type √ 1 i ωdt + B dW (t) (P+ − P− ) − Bdt UW (t) (5.33) dUW (t) = 2 8 and d˜ ρ(f ; t) =

i√ B [P+ − P− , ρ˜(f ; t)] dW (t) + L˜B [˜ ρ(f ; t)] dt , 2

(5.34)

where B L˜B [̺] := ([(P+ − P− ) ̺, P+ − P− ] + [P+ − P− , ̺ (P+ − P− )]) 8 h i X ˜ ,̺ + 1 ([D(zk )̺ , D(zk )∗ ] + [D(zk ) , ̺D(zk )∗ ]) . −i H 2

(5.35)

k

o n The quantum dynamical semigroup exp L˜B t will be one of the main ingredients in the computations of the fluorescence spectrum; therefore, we need to study its properties and, in particular, its equilibrium state. The parameters

x, y, z γ

First of all we need to summarize all the parameters which enter the model and to introduce some shorthand notations. We already started to give all quantities in units of the natural linewidth |α|2 ; for instance, we wrote λ = αΩ eiδ λ+ in (5.20), so that Ω 2 becomes an adimensional measure of the laser intensity. Then, we define


x :=

ν − ω0 , |α|2

y := B/|α|2 ,

2

z := (ω − ω0 ) /|α| , γ˜ , γ := |α|2

73

reduced frequency,

(5.36a)

reduced laser bandwidth,

(5.36b)

reduced detuning,

(5.36c)

reduced instrumental width,

(5.36d)

It is also useful to introduce the shorthand notation γ+y q := + i(x − z) 2 and the shifted detuning parameter

q (5.37) z˜

2

z˜ := z − Ω ε ,

(5.38)

where

ε, P⊥ ε := ImhS + λ+ |P⊥ S − λ+ i ,

(5.39)

P⊥ := 1l − |1, 1; +1ih1, 1; +1| ;

(5.40)

2

here Ω ε plays the role of an intensity dependent light shift. It is also useful to define δ± := δ± (1; +1) , g± := P⊥ S ± − 1l λ+ ,

s := δ+ − δ− ,

δ± , s, g±

(5.41)

∆g := g+ − g− , (5.42) 1 1 b := (5.43) 1 + y + Ω 2 k∆gk2 + sin2 s − i z˜ + Ω 2 sin 2s , 2 4 2 2 2 + Ω 2 2 + 2y + 2Ω 2 k∆gk + Ω 2 sin2 s . Γ 2 := 1 + y + Ω 2 k∆gk

(5.44)

We have to ask |ε| < +∞, kg± k < +∞, for ∆θ ↓ 0; roughly speaking, S ± must introduce small corrections even when the norm of λ+ diverges. The equilibrium state By inserting the explicit expressions of all the quantities appearing in the time-independent Liouvillian (5.35) and by using the shorthand notations just introduced, after some computations we obtain −2

|α|

L˜B [̺] = −P+ ̺P+ − bP+ ̺P− − bP− ̺P+ +

1 X

Qm ̺Q∗m

m=−1

i 1 h + Ω ei(δ+2δ− ) Q∗1 ̺ + e−i(δ+2δ− ) ̺Q1 2 i 1 X h i(δ+2δǫ ) e Pǫ ̺Q∗1 + e−i(δ+2δǫ ) Q1 ̺Pǫ . − Ω 2 ǫ=±

(5.45)

∆g, b, Γ 2

74

ρ∞

Alberto Barchielli

n o 2 ˜ has a For |α| > 0, Ω > 0, the quantum dynamical semigroup exp Lt unique equilibrium state with support n in theolinear span of |F+ , F+ i, |F− , F− i: for any initial state ρ0 , limt→+∞ exp L˜B t [ρ0 ] = ρ∞ . We skip proofs and

computations and give the final expression for ρ∞ : hF+ , F+ |ρ∞ |F+ , F+ i = Ω 2 Re d ,

hF− , F− |ρ∞ |F− , F− i = 1 − Ω 2 Re d , (5.46a)

hF+ , F+ |ρ∞ |F− , F− i = hF− , F− |ρ∞ |F+ , F+ i = Ω exp [i (δ + 2δ− )] d , (5.46b) 1 + y + Ω 2 k∆gk2 + sin2 s + i 2˜ z − Ω 2 sin s cos s d= . (5.46c) 4˜ z2 + Γ 2 5.3 The detection scheme Heterodyne detection The spectrum of our two-level atom can be scanned by using the balanced heterodyne scheme (Section 3.5 and Figure 3). We take a monochromatic laser of frequency ν as local oscillator and consider a measuring apparatus which spans a small solid angle with the vertex in the atom and not containing the forward direction in order that the light of the stimulating laser does not hit directly the apparatus. This means that we are in the case of eqs. (3.117)– (3.132). The measurement scheme produces an output current ˆ t) = k I(ν, h;

Z

t

ˆ s) , ˜ h; e−˜γ (t−s)/2 dQ(ν,

(5.47)

0

ˆ s) is the stochastic process associated to the compatible quan˜ h; where Q(ν, tum observables i XZ t h ˆ dA† (s) + eiνs hh|z ˆ j i dAj (s) , ˆ t) = e−iνs hzj |hi (5.48) Q(ν, h; j j

0

ν ∈ R is the frequency of the local oscillator, γ e > 0 is an instrumental width, ˆ ∈ Z, khk ˆ = 1. Any information on k 6= 0 is a proportionality constant and h ˆ We the localization and on the polarization of the detector is contained in h. ˆ assume that the detector spans a small solid angle, so that h is given by (θ′ , φ′ ) ˆ ′ , φ′ ) = 1Ξp h(θ ~e(θ′ , φ′ ), |Ξ|

(5.49)

where Ξ is a small solid angle around the direction (θ, φ), Ξ ↓ {(θ, φ)}, |Ξ| ≃ sin θ dθ dφ and ~e is a complex polarization vector, |~e(θ′ , φ′ )| = 1. Moreover, we assume that the transmitted wave does not reach the detector, i.e. Ξ and the laser solid angle of (5.21) are disjoint: Ξ ∩ {θ ∈ [0, ∆θ], φ ∈ [0, 2π)} = ∅;


75

in the limit of infinitesimal angles we have simply that Ξ is a small solid angle around the direction (θ, φ) with θ > 0. In particular this gives ˆ +i = 0 . hh|λ

(5.50)

Then, the generator (3.125) of the characteristic operator associated to the ˆ s) becomes ˜ h; process Q(ν, κ2 Kt (f ; κ)[̺] = L f (t) [̺] − ̺ + iκ [Z(f ; t)̺ + ̺Z(f ; t)∗ ] , (5.51) 2 where, by eqs. (3.59), (3.63), L f (t) is the generator of the dynamics Υ (f ; t, s) and is given by eqs. (5.23)–(5.25), while Z(f ; t) is given by eq. (3.126), which now becomes X 1 X ˆ ǫ − 1l)f (t)iPǫ . (5.52) ˆ m; +1iQm + hh|(S hh|1, Z(f ; t) = eiνt α ǫ=±

m=−1

By using the explicit expression (5.18) of f (t) and the stochastic unitary operators UW (t) (5.29) we get n h io √ ˆ W (t) , Z(f ; t) = exp −i (ω − ν) t + BW (t) UW (t)∗ D(h)U (5.53) where D(·) is defined in eq. (5.30). By defining

Υ˜ (t, s)[̺] := UW (t)Υ (f ; t, s) [UW (s)∗ ̺UW (s)] UW (t)∗ ,

(5.54)

eqs. (5.32), (5.34) give dΥ˜ (t, s)[̺] =

i i h i√ h B P+ − P− , Υ˜ (t, s)[̺] dW (t) + L˜B Υ˜ (t, s)[̺] dt , (5.55) 2

with L˜B given by eq. (5.35). The power spectrum

As the power of a current is proportional to the square of the current itself, the expression ˆ = lim k1 P (ν, h) T →+∞ T

Z

0

T

i h ˆ t)2 dt Eρ0 I(ν, h;

(5.56)

is the mean output power in the long run; k1 > 0 is a suitable constat with the dimensions of a resistance, it is independent of ν, but it can depend on the ˆ gives the other features of the detection apparatus. As a function of ν, P (ν, h) ˆ power spectrum observed in the “channel h”; in the case of the choice (5.49)

76

Alberto Barchielli

it is the spectrum observed around the direction (θ, φ) and with polarization ~e. Let us pospone the computations of the mean power and let us start by giving and discussing its final expression: 2 2 ˆ = k k1 + 4πk k1 Σ(ν; h) ˆ , P (ν, h) γ˜ γ˜

ˆ = Σ(ν; h)

1 2π

Z

+∞

0

(5.57)

n h io ˆ ∗ eK1 t D(h)ρ ˆ ∞ dt e−[(γe+B)/2+i(ν−ω)]t Tr D(h) + c.c.,

K1 [̺] = L˜B [̺] − BP+ ̺P− + BP− ̺P+ ,

(5.58) (5.59)

c.c. means “complex conjugated”. • The term k2 k1 γ˜ is independent of ν and, for this reason, can be seen as a white noise contribution to the power spectrum. It is due to the detection scheme, but it cannot be eliminated; its origin can be traced to the canonical commutation relations of the fields and, so, it is of a quantum origin. It is known as shot noise. ˆ is positive because it is the expectation of the • The mean power P (ν, h) ˆ is possquare of a real quantity, but it can be shown that also Σ(ν; h) ˆ itive. Therefore, Σ(ν; h) can be separated from the shot noise and can ˆ To prove be interpreted as the fluorescence spectrum in the channel h. ˆ the positivity of Σ(ν; h) one needs to go back to expressions in which the quantum fields appear explicitly; see Section 3.1.3 of [28]. • The normalization of Σ(ν; h) in (5.57) has been chosen in such a way that Z

+∞

−∞

n o ˆ dν = Tr D(h) ˆ ∗ D(h)ρ ˆ ∞ , Σ(ν; h)

(5.60)

With this choice, the total strength of the spectrum is the asymptotic rate ˆ indeed, we have of emission of photons in the “channel h”; Z +∞ ˆ dν = lim 1 TrH⊗Γ N out (Pˆ ; T )s , Σ(ν; h) (5.61) h T →+∞ T −∞

ˆ Also for this result we refer where Phˆ is the orthogonal projection on h. to Section 3.1.3 of [28]. • The state ρ∞ is the unique equilibrium state of the quantum dynamical semigroup exp(L˜B t) and it is given in eqs. (5.46). • In quantum optics it is often stated that the emission spectrum is the Fourier transform of the two-times quantum correlation function of the dipole operator. This is indeed the structure appearing in eq. (5.58) if we


77

interprete the operator D(·), defined in eq. (5.30), as an effective dipole operator. This is reasonable because a dipole operator has to take into account not only the two levels remained in the description, but also the full structure of the atom and the operator S − 1l is indeed a track of this structure. Let us note that the effective dipole operator appears also in the Liouvillian L˜B and, so it contributes to the spectrum also through ρ∞ and K1 . • The semigroup exp[K1 t] is trace preserving, but not positivity preserving, while L˜B is a bona-fide Liouvillian, because it can be written in the Lindblad form; this peculiar structure of the quantum correlation function appearing in (5.58), while not explicitly formulated, was already found in [68]. By putting in evidence the terms with B we can write K1 [̺] −

B B B ̺ = L˜0 [̺] − P+ ̺P+ − P− ̺P− + BP+ ̺P− . 2 2 2

(5.62)

Let us sketch now the computations which bring to eqs. (5.57)–(5.59). First step. By particularizing the formula for the second moments (3.132) to our case we get Z s2 Z t i k2 2 −˜ γt 2 ˆ Eρ0 I(ν, h; t) = ds2 e−˜γ (2t−s1 −s2 )/2 ds1 1−e + 2k γ˜ 0 0 io n h h √ ˆ ∗ Υ˜ (s1 , s2 ) ρ˜(f ; s2 )D(h) ˆ ∗ × Ec ei[(ω−ν)(s1 +s2 )+ B(W (s1 )+W (s2 ))] Tr D(h) n h io √ ˆ ∗ Υ˜ (s1 , s2 ) D(h)˜ ˆ ρ(f ; s2 ) + ei[(ω−ν)(s1 −s2 )+ B(W (s1 )−W (s2 ))] Tr D(h) i + c.c. . (5.63) h

Second step. The dynamics Υ˜ (t, s) depends on the Wiener process only through the increments W (τ ) − W (s), s ≤ τ ≤ t, and, so, the conditional i√B(W (t)−W (s)) c Υ˜ (t, s) Fsc is non random and coincides with expectation E e √ the expectation Ec ei B(W (t)−W (s)) Υ˜ (t, s) . Recall that Fsc is the σ-algebra generated by W (τ ), τ ∈ [0, s]; see S3’ for this notation. By eq. (5.55) and the classical Itô’s formula we get

d e

√ i B(W (t)−W (s))

√ 1 ˜ [P+ − P− , ·] + 1l dW (t) Υ (t, s)[̺] = i B 2 h √ i B + K1 − 1l dt ei B(W (t)−W (s)) Υ˜ (t, s)[̺] ; (5.64) 2

this gives √ Ec ei B(W (t)−W (s)) Υ˜ (t, s) Fsc = e−B(t−s)/2 eK1 (t−s) .

Similarly, we get

(5.65)

78

Alberto Barchielli

√ Ec e2i BW (s) ρ˜(f ; s) = e−2Bs eK2 s [ρ0 ] ,

with

K2 = K1 −

B [P+ − P− , ·] , 2

(5.66) (5.67)

or, more explicitly, 3 1 K2 [̺] − 2B̺ = L˜0 [̺] − 2B P+ ̺P+ + P− ̺P− + P− ̺P+ + P+ ̺P− . 4 4 (5.68) By inserting these results into eq. (5.63) we get Z s2 Z t h i 2 ˆ t)2 = k 1 − e−˜γ t + 2k2 ds2 e−˜γ (2t−s1 −s2 )/2 ds1 Eρ0 I(ν, h; γ˜ 0 0 n n h io B ˆ ∗ eK1 (s1 −s2 ) eK2 s2 [ρ0 ]D(h) ˆ ∗ × ei(ω−ν)(s1 +s2 )− 2 (s1 +3s2 ) Tr D(h) n h io B ˆ ∗ eK1 (s1 −s2 ) D(h)e ˆ L˜B s2 [ρ0 ] + ei(ω−ν)(s1 −s2 )− 2 (s1 −s2 ) Tr D(h) o + c.c. . (5.69)

Third step. By using the new variables of integration τ = s1 − s2 , s = s2 , we get h i 2 ˆ t)2 = k 1 − e−˜γ t + 2k2 [a1 (t) + a2 (t) + a3 (t) + c.c.] , (5.70) Eρ0 I(ν, h; γ˜ Z t Z t−τ n h io τ B ˆ ∗ eK1 τ D(h)ρ ˆ ∞ , ds e−˜γ (t−s− 2 )+[i(ω−ν)− 2 ]τ Tr D(h) dτ a1 (t) := 0

0

a2 (t) :=

Z

t

dτ

0

a3 (t) :=

Z

0

(5.71a)

Z

t−τ

0

t

dτ

Z

0

t−τ

ds e−˜γ (t−s− 2 )+[i(ω−ν)− 2 ]τ n h io ˆ ∗ eK1 τ D(h)e ˆ L˜B s [ρ0 − ρ∞ ] , × Tr D(h) τ

B

ds e−˜γ (t−s− 2 )+[i(ω−ν)− 2 ]τ io h n ˆ ∗ , ˆ ∗ eK1 τ eK3 (ν)s [ρ0 ]D(h) × Tr D(h) τ

(5.71b)

B

K3 (ν) := 2[i(ω − ν) − B] + K2 .

(5.71c) (5.71d)

The a1 term becomes Z i γ ˜ γ ˜ B 1 t h − γ˜ τ dτ e 2 − e− 2 t e− 2 (t−τ ) e[i(ω−ν)− 2 ]τ a1 (t) = γ˜ 0 n h io ˆ ∗ eK1 τ D(h)ρ ˆ ∞ × Tr D(h) (5.72)


79

and in the limit it gives Z n h io B+˜ γ 1 +∞ ˆ ∗ eK1 τ D(h)ρ ˆ ∞ . (5.73) lim a1 (t) = dτ e[i(ω−ν)− 2 ]τ Tr D(h) t→+∞ γ˜ 0 The a2 term becomes a2 (t) =

Z

0

t

τ i(ω−ν)− B ] [ ˆ ∗ 2 Tr D(h) dτ e

γ ˜ γ ˜ ˜B (t−τ )− γ˜ τ L 2 − e− 2 t e− 2 (t−τ ) ˆ e [ρ0 − ρ∞ ] ; × eK1 τ D(h) γ˜ + L˜B

(5.74)

˜

because limt→+∞ eLB t [ρ0 − ρ∞ ] = 0, we get lim a2 (t) = 0 .

(5.75)

t→+∞

Similarly, the a3 term becomes a3 (t) =

Z

t

0

B ˆ ∗ dτ e[i(ω−ν)− 2 ]τ Tr D(h) ×e

K1 τ

γ ˜

γ ˜

γ ˜

eK3 (ν)(t−τ )− 2 τ − e− 2 t e− 2 (t−τ ) ˆ ∗ [ρ0 ]D(h) γ˜ + K3 (ν)

;

(5.76)

then, at least almost everywhere in ν, 1 lim t→+∞ T

Z

0

Z γ 1 T τ i(ω−ν)− B+˜ ] [ ˆ ∗ 2 Tr D(h) a3 (t)dt = lim dτ e t→+∞ T 0 eK3 (ν)(T −τ ) − 1l 1 ˆ ∗ = 0. ◦ [ρ0 ]D(h) × eK1 τ γ˜ + K3 (ν) K3 (ν)

T

(5.77)

By inserting all these results into (5.56) we get the expression (5.57)–(5.59) for the power spectrum. 5.4 The fluorescence spectrum In order to get an explicit expression for the spectrum (5.58), we need to solve the pseudo–master equation σ(t) ˙ = K1 [σ(t)] with the initial condition ˆ σ(0) = D(h)ρ∞ . We already said that the equilibrium state ρ∞ is supported by the two extreme states |1i = |F+ , F+ i ,

|2i = |F− , F− i

(5.78)

and one can check that all the operations involved in (5.58) leave the span of |1i, |2i invariant; so, we can forget all the other states in H and we are left with formulae involving 2 × 2 matrices. Let us use the Pauli matrices

80

Alberto Barchielli

σ+ =

01 , 00

σ− =

00 , 10

with this notation we have in particular P± =

σz = 1 2

1 0 ; 0 −1

(5.79)

(1 ± σz ) and

ˆ = αΩeiδ hh|1, ˆ 1; +1i D1 + hh|g ˆ + iP+ + hh|g ˆ − iP− , D(h)

1 −iδ e σ− − ieiδ+ sin δ+ P+ − ieiδ− sin δ− P− , D1 := Ω   Ω 2 Re d Ω exp [i (δ + 2δ− )] d . ρ∞ =  Ω exp [−i (δ + 2δ− )] d 1 − Ω 2 Re d

(5.80) (5.81) (5.82)

The angular distribution of the spectrum ˆ the To obtain the angular dependence of the spectrum, let us introduce for h states h± concentrated around (θ, φ) and with right/left circular polarization, given by equation (5.49) with ~e = ~e± , where   i sin φ ∓ cos θ cos φ exp(iφ)  −i cos φ ∓ cos θ sin φ . ~e± (θ, φ) = √ (5.83) 2 ± sin θ

Then, we can introduce the two angular spectra Σ± (x; θ) :=

1 Σ(ν; h± ) , |∆Ξ|

(5.84)

which, by the cylindrical symmetry of the problem, do not depend on φ; recall that x is linked to ν by eq. (5.36a). Then, we have Z Ω 2 +∞ −qt e Tr D± (θ)∗ eK1 t [D± (θ)ρ∞ ] dt + c.c., (5.85) Σ± (x; θ) = 2π 0 r X 1 3 D± (θ) := ± (1 ± cos θ) D1 + gǫ (θ; ±)Pǫ , (5.86) 4 2π ǫ=± hh± |gǫ i . gǫ (θ; ±) := p |∆Ξ|

(5.87)

The functions gǫ (θ; ±) depend on S ǫ , but after all they are free parameters of the theory: they are square integrable θ-functions, satisfying the constraint Z π sin θ [(1 + cos θ) gǫ (θ; +) − (1 − cos θ) gǫ (θ; −)] dθ = 0 , (5.88) 0

coming from the orthogonality of gǫ to |1, 1; +1i, see (5.42). Then, by integrating over the whole solid angle, one gets the total spectrum


Σ(x) =

Z

0

π

dθ sin θ

Z

0

2π

X dφ Σ+ (x; θ) + Σ− (x; θ) = Σ(ν; hk ) ,

81

(5.89)

k

where {hk } is any c.o.n.s. in Z. When one has gǫ = 0, as in the usual case, one gets 2 3 1 ± cos θ Σ± (x; θ) = Σ(x) . (5.90) 8π 2 For gǫ 6= 0 the x and θ dependencies do not factorize. In the experiments one measures something proportional to Σ+ (x; θ) + Σ− (x; θ) for θ around π/2; this quantity fails to be proportional to Σ(x) only by the presence of some terms which we expect to be small and which are not qualitatively different from the other terms in Σ(x). So, for simplicity, we shall study only the total spectrum (5.89). The total spectrum By choosing in (5.89) a basis with h2 = k∆gk−1 ∆g , (5.91a) −1/2 h3 = k∆gk−1 kg− k2 k∆gk2 − |h∆g|g− i|2 k∆gk2 g− − h∆g|g− i∆g , (5.91b) h1 = |1, 1; +1i,

we get the final expression of the total spectrum Ω2 1 Σ(x) = v3 + ieiδ− sin δ− + iΩ 2 v1 e−is sin s d − ie−iδ− sin δ− 2π q − iΩ 2 {Re d}eis sin s + e−iδ− sin δ+ Ω 2 c1 sin s − ieis c3 1 g− g− + Ω 2 {Re d}∆g + Re d + id eis sin s u3 + iΩ 2 u1 e−is sin s + q Ω 2 2 2 2 2 + Ω h∆g|g+ i c1 − Ω d k∆gk u1 + ∆g g− + Ω d1 ∆g v1 + c.c. , q (5.92)   0 1 1 , v= 2 (K + q) 1

    Re d 0 1  1 0 , d , u = c= K +q K +q 1 d   1 −1/2 −1/2 K =  Ω 2 eis cos s b + y 0  . Ω 2 e−is cos s 0 b − y

(5.93)

(5.94)

By integrating over the reduced frequency x, we get the strength of the total spectrum [cf. (5.60)]

82

Alberto Barchielli

|α|2

Z

n Σ(x) dx = |α|2 Ω 2 {Re d} 1 + Ω 2 sin2 δ+ + 1 − Ω 2 Re d −∞ o × sin2 δ− + kg− k2 + Re d e2iδ− − 1 + Ω 2 {Re d}kg+ k2 . (5.95) +∞

Let us recall that |α|2 is the natural line width, Ω 2 is proportional to the laser intensity, z = (ω − ω0 ) |α|2 is the reduced detuning, y = B/|α|2 is the reduced laser bandwidth, Ω 2 |α|2 ε is an intensity dependent shift, x = (ν − ω0 ) /|α|2 and γ = γ e/|α|2 are the reduced frequency and the reduced instrumental width, respectively, and q = i(x − z) + (γ + y) /2, s = δ+ − δ− , 2 ∆g = g+ − g− . Let us note that ε, δ± , kg± k , hg+ |g− i are parameters linked ± to the S scattering matrices, satisfying the two constraints hg+ |g− i ≤ kg+ k kg− k , (5.96a) k∆gk = 0 ⇒ ε = 0 ; (5.96b) apart from this relation ε is an independent parameter of the model. One can check that the spectrum Σ(x) is invariant under the transformation: x → −x , δ± → −δ± ,

z → −z , ε → −ε , hg− |g+ i → hg+ |g− i.

(5.97a) (5.97b)

The case S ± = 1l Let us recall that the usual model, with only the absorption/emission process, corresponds to δ± = 0, g± = 0, ε = 0, z = z˜, s = 0. In this case we obtain Ω2 1 4Ω 2 Σ(x) = v3 d + 2v3 + Re d + c.c. , (5.98) 2π q N d=

1 + y + 2iz , 4z 2 + Γ 2

Γ 2 = (1 + y)(1 + y + 2Ω 2 ) ,

v3 = [2 + γ + y + 2i (x − z)] [1 + γ + 4y + 2i (x − 2z)] N ,

(5.99) (5.100)

N = 4Ω 2 [1 + γ + 2y + 2i(x − z)] + [2 + γ + y + 2i (x − z)]

× [1 + γ + 4y + 2i (x − 2z)] (1 + γ + 2ix) . (5.101)

Now, the spectrum Σ(x) is invariant under the transformation: x → −x ,

z → −z .

(5.102)


83

Plots Let us end by presenting some plots of the spectrum with various choices of the parameters. Let us consider only the resonant case (no detuning: z = 0) and let us take γ = 0.6 for the reduced instrumental width; let us recall the in our units the natural line width is 1. In all the plots we compare the spectrum predicted by the usual model (dashed lines), in which only the absorption/emission channel is present, with the spectrum predicted by the modified model (solid lines), in which both the absorption/emission channel and the direct scattering channel are present. The usual model is characterized by δ± = 0, kg± k2 = 0, ε = 0, while as an example of modified model we choose δ+ = −0.03, δ− = 0.13, kg+ k2 = 0.0045, kg− k2 = 0.0055, hg+ |g− i = −0.004 + i × 0.002, ε = −0.001. In Figures 8, 9, 10 we consider three laser intensities Ω 2 = 3, 28, 50 and four laser bandwidths y = 0, 0.5, 1, 4. Our choice of the parameters for the modified model is such that for a monochromatic laser in resonance (Figure 1(a)) the modified spectrum is not quantitatively too different from the usual one, but its asymmetry is clear. The differences between the two cases are enhanced by the presence of the bandwidth. In Figure 11 we have considered a stimulating laser in resonance with a very large bandwidth y = 50 and four levels of intensity: Ω 2 = 5, 15, 30, 60. The instrumentals width is again γ = 0.6. 0.18

0.18 2

Ω =3, y = 0

(a) 0.15

2

Ω =3, y = 0.5

(b) 0.15

0.12

0.12

0.09

0.09

0.06

0.06

0.03

0.03

0

0 -7

-3.5

0

3.5

7

0.18

-7

-3.5

0

3.5

7

0.18 2

Ω =3, y = 1

(c) 0.15

2

Ω =3, y = 4

(d) 0.15

0.12

0.12

0.09

0.09

0.06

0.06

0.03

0.03

0

0 -7

-3.5

0

3.5

7

-7

-3.5

0

3.5

7

Fig. 8. The spectrum Σ(x) × 100 for laser bandwidths y = 0, 0.5, 1, 4 and γ = 0.6, Ω 2 = 3, z = 0.

84

Alberto Barchielli 0.1

0.1 2

Ω =28, y = 0

(a)

0.08

0.06

0.06

0.04

0.04

0.02

0.02

0 -10

-5

0

5

2

Ω =28, y = 0.5

(b)

0.08

10

0.1

0 -10

-5

0

5

10

0.1 2

Ω =28, y = 1

(c)

0.08

0.06

0.06

0.04

0.04

0.02

0.02

0 -10

-5

0

5

2

Ω =28, y = 4

(d)

0.08

10

0 -10

-5

0

5

10

Fig. 9. The spectrum Σ(x) × 100 for laser bandwidths y = 0, 0.5, 1, 4 and γ = 0.6, Ω 2 = 27, z = 0.

0.15

0.15 2

Ω =50, y = 0

(a)

0.12

0.09

0.09

0.06

0.06

0.03

0.03

0 -15

-10

-5

0

5

10

15

0.15

0 -15

-10

-5

0

5

10

15

0.15 2

Ω =50, y = 1

(c)

0.12

0.09

0.09

0.06

0.06

0.03

0.03

-10

-5

0

5

10

2

Ω =50, y = 4

(d)

0.12

0 -15

2

Ω =50, y = 0.5

(b)

0.12

15

0 -15

-10

-5

0

5

10

15

Fig. 10. The spectrum Σ(x)×100 for laser bandwidths y = 0, 0.5, 1, 4 and γ = 0.6, Ω 2 = 50, z = 0.

Quantum Continual Measurements 0.07 0.06

0.07 2

Ω =5, y = 50

(a)

0.06

0.05

0.05

0.04

0.04

0.03

0.03

0.02

0.02

0.01

0.01

0 -15

-10

-5

0

5

10

15

0.07 0.06

2

Ω =15, y = 50

(b)

0 -15

-10

-5

0

5

10

15

0.07 2

Ω =30, y = 50

(c)

0.06

0.05

0.05

0.04

0.04

0.03

0.03

0.02

0.02

0.01

0.01

0 -15

85

-10

-5

0

5

10

15

0 -15

2

Ω =60, y = 50

(d)

-10

-5

0

5

10

15

Fig. 11. The spectrum Σ(x) × 100 for laser bandwidth y = 50 and γ = 0.6, Ω 2 = 5, 15, 30, 60, z = 0.

Acknowledgments Work supported in part by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00279, QP-Applications, and by Istituto Nazionale di Fisica Nucleare, Sezione di Milano. The author is indebted to Franco Fagnola e Matteo Gregoratti for discussions and suggestions and to Stephane Attal for the invitation to the stimu´ lating “Ecole d’été de Mathématiques 2003” in Grenoble.

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