Time-dependent spontaneous localization processes

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Physics Letters A ••• (••••) ••••••

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Physics Letters A www.elsevier.com/locate/pla

Time-dependent spontaneous localization processes F. Benatti a,b,∗ , F. Gebbia a,b a b

Department of Physics, University of Trieste, Italy INFN, Trieste, Italy

a r t i c l e

i n f o

a b s t r a c t

Article history: Available online xxxx Communicated by M.G.A. Paris

Time dependent Lindblad generators have mostly been studied for discrete variable open quantum systems. We hereby initiate the study of the complete positivity of a continuous one-dimensional quantum system subjected to time-dependent spatial localizations with back-flow of information. © 2020 Elsevier B.V. All rights reserved.

Keywords: Open quantum systems Non-Markovianity Complete positivity

1. Introduction In the case of finite dimensional open quantum systems without memory effects, the master equations ∂t ρt = L[ρt ] that describe the semigroup dynamics ρ → ρt = t [ρ ] = exp(t L)[ρ ] of their states (density matrices) ρ are characterised by the theorems of GoriniKossakowski-Sudarshan and Lindblad which fix the form of the generators L [1–4] (actually, Lindblad’s version of the theorem holds for bounded generators in any dimension):

L[ρt ] = −

i  h¯

ˆ, H



ρt +



†

K α ρt K α −

α∈ A

1 2

†

Kα Kα ,

ρt



.

(1)

They consist of a commutator with a Hamiltonian operator that by itself generates a unitary time-evolution, a noise term represented by



†

a completely positive linear operator N[ρt ] := α ∈ A K α ρt K α and by an anti-commutator which restores the probability balance. Such a structure guarantees the complete positivity of the generated time-evolution which is essential to ensure its full physical consistency. More precisely, the request that the open system dynamics t = exp(t L) preserve the positivity of all time-evolving open system states ρ is not enough. Indeed, whenever the open system is statistically coupled to any inert ancillary n-level system, dynamical maps of the form t ⊗ Idn must also preserve the positivity of all entangled states, ρent , of the compound system, thereby avoiding the appearance of negative eigenvalues in the spectrum of t ⊗ Idn [ρent ] for all t ≥ 0 [5,6]. Interestingly, for time-dependent generators Lt , the theory of non-Markovian open dynamics [7–10] has shown that one may get completely positive solutions of ∂t ρt = Lt [ρt ] even when the time-dependent generator, beside a “positive” noise term



†

K α (t ) ρt K α (t )

(2)

α∈ A

also contains a “negative” contribution of the form

−



†

K β (t ) ρt

K (t ) .

(3)

β

β∈ B

As an example, consider the one-qubit, purely dissipative master equation

∂t ρt = λt (σz ρt σz − ρt ) ,

*

ρt =

1 2

Iˆ + r t · σˆ

,

Corresponding author. E-mail addresses: [email protected] (F. Benatti), [email protected] (F. Gebbia).

https://doi.org/10.1016/j.physleta.2020.126356 0375-9601/© 2020 Elsevier B.V. All rights reserved.

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2

where λt is a smooth function of time and r t identifies the qubit density matrix σ = (σ1 , σ2 , σ3 ) is the vector of Pauli matrices. The solution is

ρ → ρt = t [ρ ] =

1 2

Iˆ + e−2κt (r1 σˆ 1 + r2 σˆ 2 ) + r3 σˆ 3 ,

ρt by a point within the unit sphere in R3 , while

t

κt :=

ds λs .

(5)

0

By inspecting the eigenvalues of the Choi-Jamiolkowski matrix

M t := t ⊗ Id[ P sym ] =

1

  Iˆ ⊗ Iˆ + e−2κt σˆ 1 ⊗ σˆ 1 − σˆ 2 ⊗ σˆ 2 + σˆ 3 ⊗ σˆ 3 ,

2

where P sym projects onto the completely symmetric state vector, one sees that κt ≥ 0 guarantees that M t is positive semi-definite and thus that t completely positive [11]. While in the standard semigroup setting of the GKSL theorem, λt = λ ≥ 0 is necessary and sufficient for t being completely positive, in the non-Markovian setting λt can be negative and yet t may still be completely positive. The positive rates λt are interpreted as the (time-dependent) frequencies at which the system loses coherence because of the presence of the environment. Such loss of coherence corresponds to information about the open system which is lost into the environment; therefore, negative “frequencies” λt < 0 are associated to a gain in coherence which is due to information back-flowing from the environment into the open system [12–15]. A similar interpretation is applicable to the “positive” and “negative” contributions to the generators in (2), respectively (3). Unfortunately no general constraints on the time-dependent Kraus operators K α (t ) in the “positive” dissipative contribution and

K β (t ) in the “negative” one are known that ensure the complete positivity of t . At present, only sufficient conditions are available that stem from concrete examples. In the following, we shall consider continuous variable so-called space localization processes in one dimension: we first generalize them by focusing on a particular non-completely positive time-dependent noise term that we explicitly solve in the case of a time-independent quadratic Hamiltonian and then discuss the complete positivity of the solutions. 2. Time-independent one-dimensional localization processes We shall focus upon a quantum system in one space-dimension described by means of the Hilbert space H of square-integrable functions on the real line and by position and momentum operators qˆ and pˆ such that [ˆq , pˆ ] = i h¯ . The states of the system are density matrices ρ acting on H and their spatial localization properties with respect to a chosen length-scale α −1/2 are witnessed by the smallness of the modulus of the entries q1 |ρ |q2 when

√

α |q1 − q2 |  1 ,

(6)

where qˆ |q = q|q . Let us consider a real function f (x) ∈ H with the following properties:

dxf (x) = 1

(7)

f (x) = f (−x) ≥ 0

(8)

R

g ( y ) :=

dx e

−i y x

f (x) ≥ 0 .

(9)

R

The linear operator on the system states given by

ρ → N[ρ ] =

√

√

dx f (x) e−i x α qˆ ρ ei x α qˆ

(10)

R

is such that

√ q1 |N[ρ ]|q2 = g ( α (q1 − q2 )) q1 |ρ |q2 .

(11)

Since g ( y ) is proportional to the Fourier transform of f (x) and f (x) is square-integrable such is g ( y ) which must then vanish when its argument becomes large; then, N[ρ ] turns out to be better spatially localized than ρ . We shall call ρ → N[ρ ] a spatial localization process and consider the associated dissipative master equation with N[ρ ] as noise term:

∂t ρt = L[ρt ] = −

i  h¯

ˆ , ρt H



− λ ρt + λ N[ρt ] .

(12)

ˆ is an at most quadratic Hamiltonian, In the above expression, λ ≥ 0 is a frequency parameter, while H

ˆ = A pˆ 2 + B pˆ qˆ + B ∗ qˆ pˆ + C qˆ 2 ; H

A, C ∈ R , B ∈ C ,

(13)

that generates the following reversible dynamics for qˆ and pˆ :



† qˆ t = Uˆ t qˆ Uˆ t = at qˆ + bt pˆ † pˆ t = Uˆ t pˆ Uˆ t = ct qˆ + dt pˆ

with

⎧ ⎪ ⎨a0 = 1, b0 = 0 c 0 = 0, d 0 = 1 ⎪ ⎩ at dt − bt ct = 1

,

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ˆ ) denotes the Hamiltonian time-evolutor. where Uˆ = exp(−i /¯h Ht Of the remaining terms in the generator L on the right hand side of master equation (12), one is the localization process and the other one, −λ ρt , is a damping term that serves to stabilize the overall probability; indeed, Tr(ρt ) = 1 for the condition (7) on f (x) yields Tr(N[ρt ]) = 1. Notice that the generator L can be recast as a standard Lindblad generator by writing



N[ρt ] =

†

K i j ,k V i j ρt V k ,

(15)

i , j ;k,

where, by means of any orthonormal basis {|ψi }i ∈N in H, V i j := |ψi ψ j | whence N is completely positive. Indeed, the entries

1

K i j ,k := √ 2π

√

√ dx f (x) ψi |e−i α x y |ψ j ψ |e−i α x y |ψk

(16)

R

make for a positive semi-definite Kossakowski matrix K = [ K i j ,k ], since, for all λi j ∈ C ,



λi j K i j ,k λk ≥ 0 ,

(17)

i , j ;k,

where z denotes complex conjugation. Once diagonalized, K allows to recast N[ρt ] in the time-independent Kraus-Stinespring form (2). As for any dynamics, in order to represent fully consistent time-evolutions, the linear maps t := exp(t L) generated by (12) must be completely positive. In the case of (12), the fulfilment of this necessary property descends from the fact that compositions of completely positive maps yield completely positive maps and from using the Trotter product formula to write

t = e−λ t lim





exp −

n→+∞

it n h¯

  n t [ H , ·] exp N[·] ,

(18)

n

where the limit can be taken with respect to the trace-norm topology with respect to which both exponentials are continuous. They are also completely positive maps, the first being a unitary trace-preserving map and the second one being an exponential series of powers of the completely positive noise term. Remark 1. Master equations with spatial localization of the type considered here belong to the class of decoherence models [16–20]. In particular, in the so-called Ghirardi-Rimini-Weber model [17] it is assumed that any quantum system is subjected to a spatial localization process that irreversibly modifies the Liouville-von Neumann master equation yielding a fundamentally dissipative equation of motion of the form

∂t ρt = −

i  h¯

ˆ , ρt H





α − λ ρt + λ π

+∞



α dq e− 2 qˆ −q

2

α qˆ −q2

ρt e− 2

(19)

.

−∞

One checks that the last term in the generator acts as

+∞

λ N(ρt ) = √

π

√

√

2 dx e−x e−i α x qˆ ρt ei α x qˆ ,

−∞

(20)

√

so that it amounts to a Gausssian spatial localization process with f (x) = exp (−x2 )/

π.

3. Time-dependent master equation In the spirit of the introduction, we now generalise the master equation (12) by considering an explicitly time-dependent spatial localization process

Nt [ρ ] =

√

√

dx f t (x) e−i x α qˆ ρ ei x α qˆ ,

(21)

R

where, for all t ≥ 0, f t (x) is a time-dependent real continuous function on the real axis with real Fourier transform,

+∞

dx e− i y x f t (x) = gt ( y ) ,

gt ( y ) :=

(22)

−∞

whence f t (x) = f t (−x), and such that

+∞ dx f t (x) = λt .

(23)

−∞

This yields a time-dependent master equation

∂t ρt = −

i  h¯

ˆ , ρt H



− λt ρt + Nt [ρt ] .

(24)

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Remark 2. Notice that the time-independent spatial localization process in (12) is represented as a random linear combination of transla√ tions in position by the amounts h¯ α x,

ρ → e−i x

√

√

α qˆ ρ ei x α qˆ ,

weighted by the probability distribution f (x). The generalization proposed in (24) amounts to allowing for a modulation of the position translations which need not be non-negative so that the dissipative and noise terms may in line of principle become associated, as briefly sketched in the Introduction, with a back-flow of information from the environment into the open quantum system. The time-independent √ master equation (19) is recovered by setting f t (u ) = λ exp(−x2 )/ π . The appearance of the unitary spatial translations makes the use of the Weyl operators most suited to seek a solution to (24). Indeed, as showed in Appendix A, one can easily solve the dual Heisenberg master equation

i 

ˆ t (q, p ) = ∂t W

h¯



ˆ,W ˆ t (q, p ) + Nt [ W ˆ t (q, p ) − λt W ˆ t (q, p )] , H

(25)

ˆ t (q, p ) denotes an initial Weyl operator where W

ˆ (q, p ) = exp W



i h¯

 ˆ ˆ (q p − p q) ,

(q, p ) ∈ R2

(26)

ˆ t (q, p ) := tT [ W ˆ (q, p )]. From (A.9) in Appendix A, one finds ˆ (q, p ) → W evolved up to time t ≥ 0 under the dual dynamics tT of t : W

ˆ t (q, p ) = G t (q, p ) Uˆ t† W ˆ (q, p ) Uˆ t W ⎛

G t (q, p ) = exp ⎝−κt +

t

√

ds g s (

(27)

⎞

α (b−s p + a−s q)⎠ , where κt :=

t

0

ds λs .

(28)

0

Finally, writing G t (q, p ) as a formal Fourier transform

G t (q, p ) =

1 2π h¯

i

dq dp e− h¯ (qp − pq) F t (q, p ) ,

where

F t (q, p ) :=

R2

1 2π h¯

i

dq dp e h¯ (qp− pq) G t (q, p ) ,

(29)

R2

and using the Weyl relation (A.4) one finally gets the dynamics for the states in the Schrödinger picture:

ρ → t [ρ ] =: ρt =

1 2π h¯

ˆ † (q, p ) . ˆ (q, p ) Uˆ t ρ Uˆ t W dq dp F t (q, p ) W †

(30)

R2

Notice however that the multiplicative function G t (q, p ) does admit Fourier transform, F t (q, p ), only in a distributional sense, namely only by defining F t (q, p ) as a linear functional acting on a suitable space of functions φ(q , p ) admitting standard Fourier transforms:

F t [φ] :=

dq dp F t (q, p ) φ(q, p ) =

R2

q, p ) := φ(

1 2π h¯

q, p ) = G t [φ]

dq dp G t (q, p ) φ(

(31)

R2

dq dp e−i (q p − p q)/¯h φ(q, p ) .

(32)

R2

In the following, we investigate which modulating functions f t (x) lead to completely positive solutions t . 4. Complete positivity In order to discuss the complete positivity of the dynamics, we shall focus upon the Heisenberg time-evolution (27). Actually, since the ˆ (q, p ) → Uˆ t† W ˆ (q, p ) Uˆ t is completely positive and the composition of completely positive maps is completely positive, unitary dynamics W we can concentrate on the maps t such that

ˆ (q, p ) → W ˆ t (q, p ) := t [ W ˆ (q, p )] = G t (q, p ) W ˆ (q, p ) . W

(33)

A necessary and sufficient condition for the complete positivity of t is that n 

μ j μi G t (qi − q j , p i − p j ) ≥ 0

i , j =1

for all n ≥ 1,

μi ∈ C and pairs (qi , p i ) ∈ R2 . The proof which follows is an adaptation from [21].

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Remark 3. The above condition expresses the so called positive definiteness of the Fourier transform G t (q, p ) of the weight function F t (q, p ) appearing in the expression of the solution ρt in (30). What we then offer below can also be taken as a practical proof of the so-called Bochner-Schwartz theorem which states that every positive definite (tempered) distribution, in our case G t (q, p ), can be obtained by Fourier transforming a positive tempered measure, in our case F t (q, p ) dq dp [23]. One indeed recognizes in the positivity of F t (q, p ) dq dp the generalization of the standard Kraus-Stinespring representation of completely positive maps as in (2) to the continuous expression embodied by (30).





Sufficiency: We need show that t ⊗ id X † X ≥ 0 for all elements of the form

X=

N  n 

μ jk Wˆ (q , p ) ⊗ | j k| ,

(35)

=1 j ,k=1

where the kets | j , 1 ≤ j ≤ n, form an orthonormal basis in Cn . The operators X are (strongly) dense in the algebra of bounded operators on H tensorized with the matrix algebra M n (C) of which the matrix units | j k| constitute a Hilbert-Schmidt orthonormal basis. Then, from (33) and (A.3) in Appendix A one computes N n     t ⊗ id X † X =

n 

μ 1 jk1 μ 2 jk2 G t (q 2 − q 1 , p 2 − p 1 ) Wˆ † (q 1 , p 1 ) Wˆ (q 2 , p 2 ) ⊗ |k1 k2 | .

(36)

j =1 1 , 2 =1 k1 ,k2 =1

If (34) holds, then for each fixed j, the quantities G t (q 2 − q 1 , p 2 − p 1 ) are the G 1 2 (t ) entries of an n × n positive matrix

 n Gt = G t (qi − q j , p i − p j )

i , j =1

(37)

,

that can be diagonalized yielding G 1 2 (t ) = H  |k := k| , one finds

n

i =1

γi (t ) g 1 i (t ) g 2 i (t ), with positive eigenvalues γi (t ) ≥ 0. Then, for all | ∈ H ⊗Cn , setting

 N n 2 n       † ˆ |t ⊗ id X X | = γi (t )  μ jk g i (t ) W (q , p ) |k  ≥ 0 .   

†

i , j =1

(38)

=1 k=1

Necessity: If t is completely positive then the right hand side of (36) is positive so that, setting

Y :=

n 

ˆ † (qk , pk ) ⊗ |k 1| , W

(39)

k =1





ˆ (−q, − p ) = 2π h¯ δ(q − q)δ( p − p ), one gets ˆ (q, p ) W and using the Weyl composition law (A.3) in Appendix A and that tr W



0 ≤ Tr Y † t [ X † X ] Y



=

N n  

n 

μ 1 jk1 μ 2 jk2 G t (q 2 − q 1 , p 2 − p 1 ) ×

j =1 1 , 2 =1 k1 ,k2 =1

 ˆ (qk1 , pk ) W ˆ † (q 1 , p 1 ) W ˆ (q 2 , p 2 ) W ˆ † (qk , pk ) ×Tr W 2 2 2 =

N n  

n 

δ((qk1 − qk2 ) − (q 1 − q 2 )) δ(( pk1 − pk2 ) − ( p 1 − p 2 )) μ 1 jk1 μ 2 jk2

(40)

j =1 1 , 2 =1 k1 ,k2 =1

× G t (q 2 − q 1 , p 2 − p 1 ) . Since the Dirac deltas are positive distributions, the arbitrariness of the complex coefficients

μ jk yields the result.

An immediate corollary of (34) is that

G t (q, p ) ≤ 1 ∀ (q, p ) ∈ R2

and

κt ≥ 0 .

(41)

Indeed choosing n = 2 and (q1 , p 1 ) = (q, p ), (q2 , p 2 ) = (0, 0) yields the positive 2 × 2 matrix (see (37))



Gt =

1 G t (q, p )

G t (q, p ) 1



(42)

√

whence the first inequality, while the second one follows by sending q, p to infinity so that the square-integrable g s (

α (b s p + a−s q)) → 0.

In order to appreciate the difficulties in certifying which modulating functions f t (x) yield damping exponentials G t (q, p ) that satisfy (34), let us consider the standard semigroup scenario by setting f t (x) = λ f (x) as in (12) so that

⎛

G t (q, p ) = exp ⎝−λ t + λ

t 0

⎞ √ ds g ( α (b−s p + a−s q))⎠ .

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6

Then, expanding the exponential of the integral in power series, one obtains

G t (q, p ) = e

∞ k  k  λ

−λ t

k!

k =0

+∞

t

=0 0

√

dx e−i α (b−s p +a−s q) f (x ) ,

ds

(44)

−∞

whence the positivity of both λ and f (x) yields n 

∞ k  k  λ

μi μ j G t (q j − qi , p j − p i ) = e−λ t

i , j =1

k =0

k!

t

=0 0

 2   n k   √  d s f (x ) ×  μ j exp −i α (b−s v p j + a−s v q j )  ≥ 0 .  j =1  v =0

From the last expression it is apparent that checking the positivity of the mean value n 

μ|Gt |μ :=

μi μ j G t (q j − qi , p j − p i )

i , j =1

for generic non-positive f t (x) becomes highly problematic. 5. Applications In order to probe whether and when μ|Gt |μ ≥ 0, in the following we limit ourselves to the case of a quantum system undergoing a ˆ = pˆ 2 /2m, subjected to a localization process with modulating function as in Remark 1, free motion determined by the Hamiltonian H

1 2 f t (x) = λt √ e−x

√

so that

π

gt (

α (b−t p + a−t q)) = λt e−α (b−t p+a−t q) , 2

(45)

where a Gaussian spatial localization is separated from a time-dependent “frequency” that need not be always positive. The chosen Hamiltonian yields at = 1 = dt , bt = t /m and ct = 0 in (14) and a damping function

t



G t (q, p ) = exp − κt +

p

α

ds λs e− 4 (q− m s)

2

(46)

.

0

α approximation of the previous expression: ⎞ ⎛  

t 2 s α p ⎠. G t (q, p ) = exp ⎝− ds λs q2 + 2 s2 − 2qp

Let us first consider the small

4

m

m

(47)

0

Through (27), this damping function provides a solution to the

ˆ t (q, p ) = ∂t W



i h¯

pˆ 2 2m



α

ˆ t (q, p ) − λt ,W

4

qˆ , qˆ ,

ρt

α → 0 approximation of (25)

 (48)

.

Then, the necessary condition for complete positivity, G t (q, p ) ≤ 1, implies the negativity of the quadratic form at the exponent:

t

κt =

t ds λs s2 ≥ 0 ,

ds λs ≥ 0 , 0

t

κt

0

⎛ ds λs s2 ≥ ⎝

0

t

⎞2 ds λs s⎠ .

(49)

0

One checks that these are necessary and sufficient conditions for the complete positivity of Gaussian maps [22]. In the more general non-Gaussian case, a sufficient condition for complete positivity is that the n × n matrix gt with entries



t g i j (t ) :=

ds λs exp −

α 4

 q j − qi −

p j − pi m

2  s

(50)

0

be positive semidefinite for all n ≥ 1 and pairs (q i , p i ) ∈ R2 . Indeed, by means of the Hadamard product of two n × n matrices, ( A , B ) → A ◦ B with entries

( A ◦ B )i j = A i j B i j ,

(51)

one recasts the entries G i j (t ) of Gt in (37) as

G i j (t ) =

∞  k =0

⎛

⎞

1 ⎜

k!

⎟ ⎝gt ◦ gt · · · ◦ gt ⎠ . " #$ % k tines

ij

(52)

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The claim is proved because the Hadamard product of positive matrices is positive itself, as one can see by reducing to one dimensional projections P = |ψ1 ψ1 | and Q = |ψ2 ψ2 |: n 

|( P ◦ Q )| =

i , j =1

 n 2     i ψ1i ψ2i  j ψ1 j ψ2 j =  i ψ1i ψ2i  ≥ 0 .  

One can thus reduce to finding those non-positive λt such that, for all n ≥ 1,

+∞

1

√

π

dx e

−x2

−∞

t

(53)

i =1

μi ∈ C and all pairs (qi , p i ) ∈ R2 ,

 n 2   √  i α x (qi − p i s/m)  ds λs  μi e  ≥0.  

(54)

i =1

0

6. Conclusions In what precedes we considered time-dependent spatial localization processes affecting one-dimensional continuous open quantum systems and explicitly solved the corresponding master equation for quadratics Hamiltonians. The real issue at stake there is the complete positivity of the dynamics when one allows for negative spatial localization frequencies and for back-flow of information from the environment into the open system. We have offered formal necessary and sufficient conditions for complete positivity that become quite concrete for Gaussian approximations. Beyond that, we could not provide explicit examples only a suitable framework where one can either numerically or analytically hopefully assess which negative “frequencies” can nevertheless lead to completely positive two parameter semigroups. Only after such a preliminary step, could one then legitimately discuss which kind of non-Markovianity pertains to which back-flow of information. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgement F.B. and F.G. acknowledge that their research has been conducted within the framework of the Trieste Institute for Theoretical Quantum Technologies. Appendix A. Solution to the time-dependent Heisenberg master equation After removing the Hamiltonian contribution to the generator of (25) by setting

ˆ t (q, p ) = Uˆ t ' W W t (q, p )Uˆ t , †

(A.1)

where U t is the unitary dynamics yielding (14), the master equation in the Heisenberg picture becomes

∂t ' W t (q, p ) = −λt ' W t (q, p ) +

+∞

ˆ (− h¯ dx f t (x) W

√

√

√

√

ˆ † (− h¯ α x b−t , h¯ α x a−t ) , α x b−t , h¯ α x a−t ) ' W t (q, p ) W

(A.2)

−∞

√

ˆ (0, h¯ with exp(−i α x qˆ ) = W gebra such that

√

√

√

√

α x) whence Wˆ (− h¯ α x b−t , h¯ α x a−t ) = Uˆ t Wˆ (0, h¯ α x) Uˆ t† . Since the Weyl operators constitute an al-

ˆ (q1 , p 1 ) W ˆ (q1 + q, p 1 + p ) ˆ (q, p ) = e−i (q1 p − q p 1 )/(2h¯ ) W W ˆ (q1 , p 1 ) W ˆ (q, p ) W ˆ (q1 , p 1 ) = e W †

−i (q1 p − p 1 q)/¯h

(A.3)

ˆ (q, p ) , W

(A.4)

we can look for a solution that is a linear combination of Weyl operators of the form

+∞

ˆ (q, p ) . dq dp F t (q, p ) W

' W t (q, p ) =

qp

(A.5)

−∞

Then using (A.4) and f t (x) = f t (−x) one gets:



+∞ dq dp −∞



qp ∂t F t (q, p )

+

qp λt F t (q, p )



+∞ −

dx f t (x) e

√ −i α x(b−t q+a−t p )

 qp F t (q, p )

ˆ (q, p ) = 0 . W

(A.6)

−∞

ˆ (−q, − p ) = 2π h¯ δ(q − q)δ( p − p ), one gets the following differential equation for the unknown function F t (q, p ), ˆ (q, p ) W Using that tr W qp

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AID:126356 /SCO Doctopic: Quantum physics

[m5G; v1.283; Prn:28/02/2020; 8:53] P.8 (1-8)

F. Benatti, F. Gebbia / Physics Letters A ••• (••••) ••••••

8

qp ∂t F t (q, p )

+∞



√

dx f t (x) 1 − e−i α x (b−t q+a−t p )

=−



qp

F t (q, p ) ,

(A.7)

−∞ qp

with initial condition F t =0 (q, p ) = δ(q − q)δ( p − p ). Then,

 qp F t (q, p )

= exp

t −

+∞ ds

0



dx f s (x) 1 − e

√ −i α x (b−s p +a−s q)

( qp

F 0 (q, p ) .

(A.8)

−∞

Once the previous expression is substituted into (A.5), one finally finds (see (22))

 ˆ (q, p ) , ' W t (q, p ) = G t (q, p ) W

G t (q, p ) = exp

t

t ds λs +

− 0

( √ ds g s ( α (b−s p + a−s q) .

(A.9)

0

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22]

A. Gorini, A. Kossakowski, E.C.G. Sudarshan, J. Math. Phys. 17 (1976) 821. G. Lindblad, Commun. Math. Phys. 48 (1976) 119. R. Alicki, K. Lendi, Quantum Dynamical Semigroups and Applications, Springer, Berlin, 1987. H.-P. Breuer, F. Petruccione, The Theory of Open Quantum Systems, Oxford University Press, Oxford, 2007. F. Benatti, R. Floreanini, Int. J. Mod. Phys. B 19 (2005) 3063. F. Benatti, R. Floreanini, R. Romano, J. Phys. A, Math. Gen. 35 (2002) L351. ´ D. Chru´scinski, A. Kossakowski, J. Phys. B, At. Mol. Opt. Phys. 45 (2012) 154002. ´ D. Chru´scinski, S. Maniscalco, Phys. Rev. Lett. 112 (2014) 120404. ´ D. Chru´scinski, F.A. Wudarski, Phys. Lett. A 377 (2013) 1425. ´ D. Chru´scinski, F.A. Wudarski, Phys. Rev. A 91 (2015) 012104. R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys. 81 (2009) 865. E. Laine, H. Breuer, J. Piilo, Phys. Rev. A 81 (2010) 062115. Á. Rivas, S.F. Huelga, M.B. Plenio, Rep. Prog. Phys. 77 (2014) 9. H. Breuer, E. Laine, J. Piilo, Phys. Rev. Lett. 103 (2009) 210401. H.-P. Breuer, E.-M. Laine, J. Piilo, B. Vacchini, Rev. Mod. Phys. 88 (2016) 021002. E. Joos, H.D. Zeh, Z. Phys. B, Condens. Matter 59 (1985) 223. G.C. Ghirardi, A. Rimini, T. Weber, Phys. Rev. D 34 (1986) 470. M.R. Gallis, G.N. Fleming, Phys. Rev. A 42 (1990) 38. B. Vacchini, K. Hornberger, Phys. Rep. 478 (2009) 71. M. Schlosshauer, Phys. Rep. 831 (2019) 1. B. Demoen, P. Vanheuswijn, A. Verbeure, Lett. Math. Phys. 2 (1997) 161. J. Eisert, M.M. Wolf, Gaussian quantum channels, in: N.J. Cerf, G. Leuchs, E.S. Polzik (Eds.), Quantum Information with Continuous Variables of Atoms and Light, Imperial College Press, 2007. [23] M. Gel’fand, N.Y. Vilenkin, Generalized Functions, vol. 4, Academic Press, 1964.