Infinite Dimensional Analysis, Quantum Probability and Related Topics c World Scientific Publishing Company
ON THE ASYMPTOTIC BEHAVIOUR OF SOME STOCHASTIC DIFFERENTIAL EQUATIONS FOR QUANTUM STATES
ALBERTO BARCHIELLI and ANNA M. PAGANONI Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy
[email protected] [email protected]
Received In this article we study the long time behaviour of a class of stochastic differential equations introduced in the theory of measurements continuous in time for quantum open systems. Such equations give the time evolution of the a posteriori states for a system underlying a continual measurement. First of all we give conditions for the equation to preserve pure states and then, in the case of a finite-dimensional Hilbert space, we obtain sufficient conditions from which the stochastic equation for a posteriori states is ensured to map, for t → +∞, mixed states into pure ones. Finally we study existence and uniqueness of an invariant measure for the equations which preserve pure states. We give a general theorem for the purely diffusive case, again in the finitedimensional case; then we apply it to some physical examples. For the purely jump case, an example is discussed in which the invariant measure exists and is unique. Keywords: Stochastic differential equations for quantum states, invariant measure, quantum open system, pure state, a priori and a posteriori states, linear entropy, quantum theory of measurements continuous in time.
1. Introduction The introduction10–12 of ideas of the classical filtering theory in the theory of measurements continuous in time in quantum mechanics4, 15 (continual measurements) gave rise to an increasing interest in the study of structural properties of a class of classical stochastic differential equations (SDE’s). The theory of continual measurements had, in the last years, some mathematical developments in connections with three different fields: with quantum stochastic calculus,5, 8 with semigroups of operators,3, 6 with SDE’s.2, 9, 13 In Ref. 9 we studied the generality of the formulation of continual measurements based on classical SDE’s and we compared it with the one based on the theory of semigroups. We obtained the class of stochastic differential equations linked to the most general semigroup of probability operators, obtained in Ref. 3, which characterizes, under some technical restrictions, the most general quantum continual measurement process. In this paper we want to study the asymptotic behaviour of a subclass of these
A. Barchielli and A. M. Paganoni
SDE’s: the time-dependence and the jump part in the stochastic equation are simpler than in the general case and some results are proved only in the finitedimensional case. Some other asymptotic results for this kind of systems are given in Refs. 20–22. The stochastic equation we study is a sort of filtering equation determining the state to be attributed to the quantum system at time t (the a posteriori state) given the output of the measurement up to time t. First we give some structural conditions from which such an equation is ensured to map mixed states into pure states for large times; then we find some conditions to ensure the existence and the uniqueness of an invariant measure for the semigroups associated to these SDE’s. The last problem involves the study of the behaviour of diffusions bounded to a compact manifold embedded into RN and needs to apply some techniques of control theory. We also give some examples of physical system in which we can verify the hypotheses which ensure the existence and the uniqueness of the invariant measure, both in the purely diffusive and in the counting case. Let us denote by B(H) the space of bounded linear operators on the separable complex Hilbert space H, by T (H) the trace-class on H, i.e. T (H) = ρ ∈ B(H) : √ Tr { ρ∗ ρ} < ∞ , and let us set hρ, ai = Tr {ρ∗ a}, ρ ∈ T (H), a ∈ B(H). Let us recall that with state we mean a trace-class operator such that ρ = ρ∗ ,
ρ ≥ 0,
hρ, 1li ≡ Tr{ρ} = 1 ;
we denote by S(H) the set of all states on H.23 Let H, Lj , Sh , j, h = 1, 2, . . . , be bounded operators on H such that H = H ∗ , ∞ ∞ X X Sh∗ Sh are strongly convergent in B(H). Let Jk be a bounded linear L∗j Lj and j=1
h=1
map on T (H) such that its adjoint Jk∗ is a normal, completely positive map on B(H) ∞ X and Jk∗ [1l] is strongly convergent to a bounded operator. Then, we introduce k=1
the following operators on T (H): L0 [ρ]
=
−i[H, ρ] +
∞ n X j=1
o 1 Lj ρL∗j − (ρL∗j Lj + L∗j Lj ρ) 2
∞ n o X 1 + Jk [ρ] − (ρJk∗ [1l] + Jk∗ [1l]ρ) , 2 k=1
L1 [ρ]
=
L
=
∞ n o X 1 Sh ρSh∗ − (ρSh∗ Sh + Sh∗ Sh ρ) , 2
(1.1)
h=1 0
L + L1 .
The adjoint operators of L, L0 , L1 are generators of norm-continuous quantum dynamical semigroups.17, 25 Let us now consider the following stochastic differential equation for trace-class
Asymptotic Behaviour of SDE’s for Quantum States
operators in S(H): dρt = L[ρt− ]dt +
∞ h i X e j (t)ρt− + ρt− L e ∗ (t) − mj (t)ρt− dWj (t) + L j j=1
+
∞ X 1 Jk [ρt− ] − ρt− dNk (t) − νk (t)d t , νk (t)
(1.2)
k=1
where e j (t) = L
mj (t) =
eiωj t Lj , ωj ∈ R, ∗ e j (t) + L e j (t)i, hρt− , L νk (t) = hρt− , Jk∗ [1l]i.
The processes Wj (t) are independent standard Wiener processes, the Nk (t) are counting processes with stochastic intensity νk (t) dt. The sum in the jump term is only on the set where the stochastic intensity νk (t) is different from zero. The natural filtration generated by the processes Wj (t) and Nk (t) is understood. With all the assumptions above Eq. (1.2) turns out to be a particular case of the general equation which describes the evolution of the family of the a posteriori states.2, 9 The interpretation of Eq. (1.2) is as follows. A quantum system described in the Hilbert space H is taken under continual observation; the output of the Z t
observation is given by the processes Nk (t) and Wj (t) +
mj (s)ds. Then, the
0
conditional state to be attributed to the system at time t, given a certain realization of the output up to time t, satisfies Eq. (1.2). As already noticed, the form of Eq. (1.2) is not the most general one, but it contains for instance the main detection schemes found in quantum optics;8 also the chosen time-dependence is natural for some systems typical of quantum optics under the so called heterodyne/homodyne detection scheme. Let us recall also that, if we set hηt , ai := E[hρt , ai],
∀a ∈ B(H) ,
(1.3)
then the state ηt satisfies the master equation dηt = L[ηt ]. dt
(1.4)
The state ηt is the state to be attributed to the system at time t if the output of the measurement is not taken into account or not known and can be called the a priori state. First of all we want to show that under some conditions Eq. (1.2) preserves pure states. We recall that in the convex set S(H) the pure states are the onedimensional projections and that a measure of “purity” of a state ρ is the so called linear entropy Tr{ρ(1l − ρ)}. This quantity always belongs to the interval [0, 1), and it is 0 if and only if ρ is a pure state. In the problem we are studying we shall consider the linear entropy g(t) and the mean linear entropy G(t) of the solution of (1.2) defined by the position g(t) := hρt , 1l − ρt i ≡ Tr {ρt (1l − ρt )} ≡ 1 − Tr ρt2 , G(t) := E[g(t)].
A. Barchielli and A. M. Paganoni
For every random statistical operator ρt we have 0 ≤ G(t) < 1; moreover, G(t) = 0 if and only if ρt is almost surely (a.s.) a pure state. So the study of the behaviour of the linear entropy is a way to analyze whether the SDE satisfied by the family of a posteriori states preserves pure states or not. Let us observe that, by considering the embedding of the trace-class into the space of the Hilbert-Schmidt operators, the process ρt can be viewed as an Hilbertspace valued semimartingale (cf. Ref. 26 Def. 23.7). Then, it is possible (cf. Ref. 26 Theor. 27.2 and Ref. 14 § 4.5) to apply Ito’s formula to the linear entropy g(t) and to the mean linear entropy G(t) as done in Proposition 1.1 and Theorem 2.1. Proposition 1.1 Equation (1.2) is ensured to preserve pure states, in the sense that ρt is a.s. a pure state for every a.s. pure initial condition, if and only if L1 = 0 Jk [ρ] and ρk := is a pure state for every k and for every pure state ρ. Tr {Jk [ρ]} Proof. By applying Ito’s formula to the mean linear entropy we obtain d G(t) = −2E hρt− , L1 [ρt− ]i dt ∞ 2 X √ e e ∗j (t) − mj (t) √ρt− − E Tr ρt− Lj (t) + L + j=1
+
∞ X
k=1
E νk (t)hρkt− , 1l − ρkt− i + Tr 2ρt2− − ρt− Jk∗ [1l] − νk (t)ρt2− .
(1.5)
Let us rewrite the first term of the derivative of G(t) in a more convenient way: − 2E hρt− , L1 [ρt− ]i = X E hρt2− , Sh∗ (1l − ρt− )Sh i − hρt− (1l − ρt− ) , Sh∗ ρt− Sh i . =2
(1.6)
h
If ρt− is almost surely a pure state, an easy calculation shows that the second term in (1.5), the second term in (1.6) and the last part of the third term in (1.5) vanish, so that ∞ ∞ X X d G(t) = 2 E hρt− , Sh∗ (1l − ρt− )Sh i + E νk (t)hρkt− , 1l − ρkt− i . dt h=1
(1.7)
k=1
Let us observe that (1.7) is the sum of two positive terms, so that if we want d G(t) = 0 for every statistical operator which is almost surely a pure state, it dt must be (1l − P )Sh P = 0 for every h and for every monodimensional projection P , that is Sh must be a multiple of the identity, and ρk must be a pure state for every k. Remark 1.1 If we ask that all the Jk preserve pure states, in order that Eq. (1.2) have the same property, then we can say more about the structure of the maps Jk .
Asymptotic Behaviour of SDE’s for Quantum States
By the results of Kraus,23 § 3, and Ozawa,27 Theor. 4.2, there exists a partition A1 , A2 of the integer numbers such that we can write for some Rk ∈ B(H) and for some monodimensional projection Pk Rk ρRk∗ , k ∈ A1 , Jk [ρ] = (1.8) hρ, J ∗ [1l]iP , k ∈ A . k 2 k 2. From Mixed to Pure States Let us take Eq. (1.2) under the hypotheses of Proposition 1.1 to guarantee that the equation preserve pure states; we want to study if it is possible to assure also that (1.2) map asymptotically mixed states into pure ones. Some examples of this behaviour in the case of linear systems are given in Ref. 16. Theorem 2.1 Let Eq. (1.2) preserve pure states and let H be finite-dimensional. If for every time t it does not exist a bidimensional projection Pt such that, ∀j, k, ( e j (t) + L e ∗ (t))Pt = zj (t)Pt Pt (L j (2.1) ∗ Pt Jk [1l]Pt = qk (t)Pt for some complex numbers zj (t) and qk (t), then Eq. (1.2) maps asymptotically, for t → ∞, mixed states into pure ones, in the sense that for every initial condition the linear entropy vanishes for long times: lim hρt , 1l − ρt i = 0,
t→∞
a.s.
(2.2)
Proof. By taking into account the results of Proposition 1.1 and Remark 1.1 and by applying Ito’s formula to the linear entropy g(t), we obtain g(t) = g(0) − −
∞ Z X
k=1
[0,t]
Z
t
0
y(s)ds − 2
∞ Z X j=1
t 0
e j (s) + L e ∗j (s) − mj (s)idWj (s) hρs2− , L
1 − hJk [ρs− ], Jk [ρs− ]i − 1 − g(s ) dNk (s) − νk (s)ds νk (s)2
(2.3)
with y(t) =
∞ X j=1
Tr
√
2 e j (t) + L e ∗j (t) − mj (t) √ρt− ρt − L
+
2 ) Jk∗ [1l] √ + νk (t) Tr ρt − ρt − + − 1l νk (t) k=1 ( √ ) √ X ρt− Jk∗ [1l] ρt− 2 + νk (t) 1 − Tr . νk (t) ∞ X
(
k∈A2
√
(2.4)
A. Barchielli and A. M. Paganoni
Apparently, y(t) is non-negative; then, from Eq. (2.3) g(t) is a supermartingale and, being bounded between 0 and 1, it is almost surely convergent to some random variable g∞ . Let us study now the process y(t); if y(t) = 0, then, by positivity, we have necessarily ( √ e j (t) + L e ∗ (t) − mj (t) √ρt− = 0, ∀j, ρ t− L j (2.5) √ √ ∗ ρt− Jk [1l] − νk (t) ρt− = 0, ∀k. But this would only be possible if an orthogonal projection Pt existed such that e j (t) + L e ∗ (t))Pt = zj (t)Pt , Pt (L j
Pt Jk∗ [1l]Pt = qk (t)Pt ,
for some numbers zj (t), qk (t); then, for every ρt− such that ρt− = Pt ρt− Pt Eqs. (2.5) hold. The state ρt− could be mixed only if Pt was at least bidimensional. Then, the condition on the non-existence of a bidimensional projection Pt given in the statement of the Theorem is a sufficient condition to guarantee y(t) > 0 unless ρt− be pure. Let us consider now the mean linear entropy G(t); by bounded convergence we have lim G(t) = E g∞ (2.6) t→∞
and by Eq. (2.3)
0 ≤ G(t) = G(0) −
Z
0
t
E y(s) ds < 1.
(2.7)
Moreover, by the previous discussion, E y(s) = 0 only if ρs− is a.s. a pure state; but if for a certain s, ρs is a.s. pure, then G(s) = 0 and by (2.7) it remains zero for all t > s and the a posteriori state remains pure. Also if lim G(t) = 0, the t→∞
statement of the Theorem follows immediately from (2.6). So it remains to show that we can exclude the case in which lim G(t) = K > 0. t→∞
Let us assume by contradiction that lim G(t) = K > 0; this implies, because t→∞ d of the monotonicity of the function G(t) that lim G(t) = 0 and so there exists a t→∞ dt sequence of times tn such that lim
n→∞
lim
n→∞
∞ X
j=1 ∞ X
k=1
Tr Tr
p
h i 2 e j (tn ) + L e ∗ (tn ) − mj (tn ) pρ − ρt − L =0 j tn n
p
h i p 2 ∗ ρt − Jk [1l] − νk (tn ) ρt− =0 n n
a.s. (2.8)
a.s.
We prove that this implies lim hρt− , 1l − ρt− i=0 n n
n→∞
a.s.
(2.9)
Asymptotic Behaviour of SDE’s for Quantum States
Let us set e j (t) + L e ∗j (t) − mj (t), A2j (t) = L
A2j+1 (t) = Jj∗ [1l] − νj (t);
then from (2.8) we have that ∀ε > 0, ∃ n0 such that ∀n ≥ n0 ∞ X p p 2 Tr ρt − Aj (tn ) ρt− < ε. n n
(2.10)
(2.11)
j=1
Let us denote by λt the maximum eigenvalue of ρt− , and by Qt a monodimensional projection on a subspace of the eigenspace related to λt . Let us recall that, if d is the dimension of the Hilbert space H, then we have λt ≥ d1 . Let us assume λt < 1, otherwise ρt− is already a pure state. We can decompose ρt− in the following way: ρt− = λt Qt + (1 − λt )ρ⊥ t− ,
(2.12)
where ρ⊥ t− =
(1l − Qt )ρt− (1l − Qt ) . Tr {(1l − Qt )ρt− (1l − Qt )}
By using this decomposition, we deduce from (2.11) that, ∀ε > 0, ∃ n0 such that ∀n ≥ n0 ∞ X j=1
, Aj (tn )Qtn Aj (tn )i + λt2n hQtn , Aj (tn )Qtn Aj (tn )i + 2λtn (1 − λtn )hρ⊥ t− n
⊥ < ε. + (1 − λtn )2 hρ⊥ − , Aj (tn )ρ − Aj (tn )i t t
(2.13)
n
n
Because of the positivity of the elements in Eq. (2.13) we can say that definitively ∞ X j=1
hQtn , Aj (tn )Qtn Aj (tn )i
0 such that definitively (1 − λt ) > h, so that we have also ∞ X j=1
hρ⊥ , Aj (tn )Qtn Aj (tn )i < t− n
∞ X j=1
ε εd ≤ , 2λtn (1 − λtn ) 2h
(2.15)
ε ε ≤ 2. 2 (1 − λtn ) h
(2.16)
, Aj (tn )ρ⊥ A (t )i < hρ⊥ t− j n t− n
n
Now, let us denote by µt the maximum eigenvalue of ρ⊥ t− , and by St a monodimensional projection on a subspace of the eigenspace related to µt . With the same argument we deduced (2.14), we can prove from (2.16) that definitively ∞ X j=1
hStn , Aj (tn )Stn Aj (tn )i
0,
Ω := 2|hα|λi| > 0.
(3.27)
If we compare (3.25) and (3.26) with (3.13) and (3.14) respectively we have kj1 = −qj2 = Rehej |αi, kj2 = qj1 = Imhej |αi, kj3 = qj3 = 0,
ω1 = 2 Imhα|λi, ω2 = 2 Rehα|λi, ω3 = −∆ω.
(3.28)
2 Let us observe ω1P +ω22 = Ω2 and that the condition of Theorem 2.1 is satisfied, P∞ that ∞ 2 indeed j=1 |kj | = j=1 |q j |2 = kαk2 > 0. We assume also that:
(A) the vectors kj are not all parallel among them;
this requirement will be used in the following. Moreover, from Eqs. (3.22) and (3.23), we have on the submanifold M A1j (x)
=
A2j (x)
=
A3j (x)
A10 (x) A20 (x) A30 (x)
=
=
=
=
kj1 (1 − x12 + x3 ) − kj2 x1 x2 ,
−kj1 x1 x2 + kj2 (1 − x22 + x3 ), −(1 +
(ω × x)1 + (ω × x)2 + (ω × x)3 +
∞ X j=1
∞ X j=1 ∞ X j=1
x3 )(kj1 x1
+
(3.29)
kj2 x2 ),
(kj · x)A1j (x) +
kαk2 x1 x3 , 2
(kj · x)A2j (x) +
kαk2 x2 x3 , 2
(kj · x)A3j (x) +
kαk2 2 (x3 − 1). 2
(3.30)
As stated, we have to study the rank of the diffusion matrix σhl ; an easy calculation shows that the rank is zero only in the point (0, 0, −1), while it is equal to one only in the points x such that for every i, j we have Aj × x · Ai = 0. By straightforward computations we obtain Aj × x · Ai = (1 + x3 )2 (ki1 kj2 − ki2 kj1 ) and we see that there would exist such a point x only if the vectors kj were all parallel among them and we have excluded this case in assumption (A). So, in order to apply Theorem 3.2, we have to study the Lie algebra only in the point x0 := (0, 0, −1). First of all we obtain A(x0 ) = (−ω2 , ω1 , 0) and {Aj , A0 }(x0 ) = 0, ∀j. So in order to verify the hypotheses of Theorem 3.2 we have
A. Barchielli and A. M. Paganoni
to compute also A0 , {A0 , Aj } (x0 ) and to check whether this in vector is linearly dependent of A0 (x0 ). This is true because it turns out that A0 , {A0 , Aj } (x0 )× A0 (x0 ) = Ω2 kj · ω and these quantities cannot be all vanishing because of as3
sumption (A). Therefore the Lie algebra is full, Theorem 3.2 can be applied and the unique invariant measure µ has Supp µ = M . In Ref. 29 the problem of invariant measures for a two-level atom is studied by means of numerical simulations. They have ω = Ω, 0, −∆ω , which is always possible by a choice of phases in (3.28); however, all the simulations are done for ∆ω = 0. The models they consider are a heterodyne detection scheme, with L1 = √1 kαkσ− , L2 = √i kαkσ− , Lj = 0 for j ≥ 3, and a homodyne detection scheme, 2 2 with L1 = kαke−iφ σ− and Lj = 0 for j ≥ 2. By (3.28) the hetherodyne scheme corresponds to 1 0 kαk kαk 0 , 1 , k1 = √ k2 = √ kj = 0 for j ≥ 3 . 2 0 2 0
In this case all the conditions from (3.27) to (A) are satisfied and an unique invariant measure exists and its support is the whole manifold M . The numerical simulation29 gives indeed an idea of a measure spread out on the whole M . The homodyne scheme with L1 = kαke−iφ σ− and Lj = 0 for j ≥ 2 corresponds to cos φ kj = 0 for j ≥ 2 . (3.31) k1 = kαk − sin φ , 0
The generic case is out of the hypotheses of Theorem 3.2, because the diffusion matrix is degenerate everywhere in M . However, the case ∆ω = 0, φ = π/2 is very interesting, because the simulation in Ref. 29 shows a tendency of the solution of the SDE to concentrate in x1 = 0. Indeed, we can prove that the circumference x22 + x32 = 1 is an invariant submanifold of M and, by differentiating x12 , one sees that x1 goes to zero for t → ∞. In the circle the diffusion matrix has rank 1 everywhere but in (0, 0, −1), where A0 is different from zero; therefore, Theorem 3.2 applies to this submanifold and also in this last case there exists a unique invariant measure which is concentrated on the circumference x22 + x32 = 1, x1 = 0. The case (3.31) can be handled with Theorem 3.3. In this case the solution of Eq. (3.10) can be easily computed via Eq. (3.11) and one that the hypotheses obtains 0 0 of Theorem 3.3 are satisfied with Γ = ∅, x0 = ρo = . This means that there 0 1 exists a unique invariant probability measure µ with support M0 implicitly given in Theorem 3.3. 4. Invariant Measure: the Counting Case As an example of an equation of jump type for the a posteriori states we take again the model of two-level atom introduced in the previous section, but now we
Asymptotic Behaviour of SDE’s for Quantum States
consider the so called direct detection scheme.1 Now we have Eqs. (1.1) and (1.2) with Sh = 0, Lj = 0, H given by Eq. (3.26) and Jk [ρ] = |hek |αi|2 σ− ρσ+ .
(4.1)
Then,Eqs. (1.1) and (1.2) reduce to dρt =
−i [H, ρt− ] −
where N (t) =
P
1 kαk2 (ρt− σ+ σ− + σ+ σ− ρt− ) + ν(t)ρt− 2
dt
+ (σ− σ+ − ρt− ) dN (t) , k
(4.2)
Nk (t) is a counting process of stochastic intensity ν(t) = kαk2 hρt− , σ+ σ− i .
The meaning of Eq. (4.2) is the following. If at time t the counting processes N presents a jump, then dN (t) = 1, and the terms containing dt are negligible and the system jumps into the state σ− σ+ , which is the projection onto the ground state, independently of the state at the jump time. In between two jumps dN (t) = 0 and ρt satisfies the deterministic equation obtained from (4.2) by suppressing the term with dN (t); the initial point after a jump is the ground state. By using the representation on the Bloch sphere introduced in the previous section, such a deterministic equation becomes dx(t) = B x(t) , dt
x(0) = (0, 0, −1) ,
0 1 1 B(x = ω × x + kαk2 x3 x − kαk2 0 ; 2 2 1
(4.3)
(4.4)
in terms of the solution of this equation, the stochastic intensity of the counting process N is given by ν(t) =
1 kαk2 [1 + x3 (t − t¯)] , 2
(4.5)
where t¯ is the time of the last jump. Note that N (t) turns out to be a renewal counting process.1 So, there exists a unique invariant measure whose support is the trajectory determined by the solution of Eq. (4.3). The finer structure of the invariant measure depends on the velocity B and on the intensity ν(t). Something more can be said about the deterministic system (4.3). As already said, without loss of generality we can take ω = (Ω, 0, −∆ω), Ω > 0; moreover, we set τ = kαk2 t, η = Ω/kαk2 , δ = ∆ω/kαk2 . Then, Eqs. (4.3), (4.4) give dx(τ ) = V (x(τ )), dτ
x(0) = (0, 0, −1),
(4.6)
A. Barchielli and A. M. Paganoni
V1 (x) = V2 (x) = V3 (x) =
1 δx2 + x1 x3 , 2 1 −δx1 − ηx3 + x2 x3 , 2 1 ηx2 − (1 − x32 ). 2
(4.7)
Let us start by the case δ = 0, for which it turns out immediately that x1 (τ ) = 0, ∀τ . Then, the motion is bound on the circumference x22 + x32 = 1 and has a tangential velocity v(τ ) = −x3 (τ )V2 (x(τ )) + x2 (τ )V3 (x(τ )) = η − 21 x2 (τ ); v(τ ) > 0 means counterclockwise motion. Therefore, for η > 21 we have v(τ ) > 0, ∀τ , and the support of the invariant measure is the whole circumference, while for η ≤ 12 the support of the p invariant measure reduces to the arc of circumference from (0, 0, −1) to (0, 2η, − 1 − 4η 2 ). For δ 6= 0 the trajectory determined by (4.6) is more complicated. One can find the critical points on M , satisfying V (x) = 0, and analyze on the tangent plane the linearized system around them. It turns out that there is always a unique stable e, attractor in x x e1
=
x e2
=
x e3
=
4ηδe x3 , x e32 + 4δ 2 2ηe x32 , x e32 + 4δ 2 v s u 2 u1 1 t − − 2η 2 − 2δ 2 + − 2η 2 − 2δ 2 + 4δ 2 ≡ −4 Re λ, 2 2 −
(4.8)
p e attracts also the where λ = 41 1 − 4η 2 − 4δ 2 − 4iδ with Re λ > 0. The point x trajectory starting from (0, 0, −1). Indeed, it is known that there always exists a linear equation equivalent to Eq. (1.2)1, 2, 9, 10 and the same is true for Eq. (4.6).1 In fact it is easy to check that, if we set x1 (τ ) = x2 (τ ) = x3 (τ ) =
2 Re u1 (τ )u2 (τ ) , |u1 (τ )|2 + |u2 (τ )|2 2 Im u1 (τ )u2 (τ ) , |u1 (τ )|2 + |u2 (τ )|2 |u1 (τ )|2 − |u2 (τ )|2 , |u1 (τ )|2 + |u2 (τ )|2
(4.9)
where the complex vector (u1 , u2 ) satisfies the linear differential equation 1 i u˙ 1 (τ ) = − (1 − 2iδ)u1 (τ ) − ηu2 (τ ) 4 2 1 i u˙ 2 (τ ) = (1 − 2iδ)u2 (τ ) − ηu1 (τ ) 4 2
(4.10)
with the initial condition u1 (0) = 0, u2 (0) = 1, then x(τ ) satisfies Eq. (4.6) with initial condition (0, 0, −1). This means that Eq. (4.6) is analytically solvable and
Asymptotic Behaviour of SDE’s for Quantum States
e is the limit point of the trajectory we are interested that for δ 6= 0 the point x in. Note that the two characteristic roots of the system (4.10) are ±λ, where λ is the complex parameter introduced after Eq. (4.8). Numerical simulations, together with a graphical representation of the invariant measure are given in Ref. 29 Let us end by discussing the connections between the invariant measure and the equilibrium state of the master equation (1.4). The following considerations apply both to the diffusive and to the jump case; we consider Eq. (1.2) in the time homogeneous case, i.e. ωj = 0, ∀j, and ρt is a generic solution of this equation. Let us assume that H is finite dimensional and that there exists a unique invariant probability measure µ and that Supp µ ⊂ M . By definition of invariant measure we have Z Z Ex [f (x(t)]µ(dx) = f (x)µ(dx) (4.11) M
M
for every bounded measurable complex function f on M ; Ex is the expectation in the case the initial condition for the process is x. Moreover, by the uniqueness of µ we have ergodicity, i.e. Z Z 1 T lim f (x(t))dt = f (x)µ(dx) a.s. (4.12) T →+∞ T 0 M By applying these two facts to the function f (x) = hρ(x), ai, where a is an arbitrary linear operator on H and ρ is given by Eq. (3.7), and by recalling that Ex [ρt ] satisfies the master equation (1.4), we obtain that the equation L[η] = 0 has a unique solution solution ηeq in S(H), given by Z ηeq = ρ(x)µ(dx); (4.13) M
moreover for every initial condition Z 1 T lim ρt dt = ηeq T →+∞ T 0
a.s.
(4.14)
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