Many unstable particles from an open quantum systems perspective
aa r X i v : . [ qu a n t - ph ] M a y Many unstable particles from an open quantum systemsperspective
Kordian Andrzej Smoli´nski ∗ Department of Theoretical Physics, Faculty of Physics and Applied Computer SciencesUniversity of L´od´z, ul. Pomorska 149/153, 90-236 L´od´z, Poland
Abstract
We postulate a master equation, written in the language of creation and annihilation operators, asa candidate for unambiguous quantum mechanical description of unstable particles. We have foundKraus representation for the evolution driven by this master equation and study its properties.Both Schr¨odinger and Heisenberg picture of the system evolution are presented. We show that theresulting time evolution leads to exponential decay law. Moreover, we analyse mixing of particleflavours and we show that it can lead to flavour oscillation phenomenon.
PACS numbers: 03.65.Yz, 03.65.Ca, 02.50.Ga ∗ [email protected] . INTRODUCTION One of important difficulties in quantum mechanical description of unstable particles isirreversibility of time-evolution. The complete system consists of decaying particle as wellas of decay products. Only this complete system undergoes unitary evolution, describedby quantum field theory. However, in many applications of quantum mechanics (e.g., inanalysis of correlations experiments) we would like to neglect the evolution of decay productsand consider the decaying particles only. This is usually achieved by introducing a non-Hermitian Hamiltonian, as it was done in the classical works of Weisskopf and Wigner[1, 2]. The non-Hermitian Hamiltonian, however, leads inevitably to the non-conservationof trace of the density operator of the system. Although such description gives decreasingprobability of detecting the particle, it does not provide unambiguous way of calculating theprobability of finding the system consisting of a few such particles in a given state after themeasurement (see e.g. [3] for a discussion of other ambiguities caused by various approachesto the description of such a system). Indeed, one needs to use probability theory ratherthan quantum mechanics for this purpose. Since the calculations of conditional probabilitiesare extremely important for analysis of correlation experiments, especially of those done forsystems of neutral kaons [4–12] or B -mesons [13–15], it would be desirable to find a quantummechanical description of decaying particles preserving unit trace and positivity [16–22] ofthe density operator for the system. Recent papers (e.g. [23, 24]) show that there is still agreat interest in the unambiguous quantum-mechanical description of neutral kaon system.This is the point where the theory of dynamical semigroups and open quantum systems[25–27] can be helpful. Let us recall (cf. [28–30]) that the dynamical semigroup in theSchr¨odinger picture is a one-parameter family of linear maps Λ ∗ t , acting on the space oftrace class operators on Hilbert space of the system, preserving for every t ≥
0: (i) positivity,(ii) trace, (iii) strong continuity and such that (iv) Λ ∗ t Λ ∗ t = Λ ∗ t + t for every t , t ≥
0. Theseproperties can be translated into Heisenberg picture as requirements for the map Λ t actingon the space of bounded operators on the Hilbert space of the system, which for every t ≥ t Λ t = Λ t + t for t , t ≥ K K system [37, 38].Here, we follow the approach presented in [35, 37, 38]. However, in these works theconsiderations were restricted to systems of at most two particles, and transition from one-particle to two-particle theory was done by means of tensor product construction. In thepresent paper we will show that it is possible to describe systems with arbitrary number ofparticles using the second quantization formalism, which is the most natural language forsystem with varying number of particles. Moreover, such approach would be an advantageif we study the behaviour of the system from uniformly moving or accelerated frame, due tothe well established transformation properties of annihilation and creation operators.The paper is organized as follows. In Sect. II, we postulate a master equation inSchr¨odinger picture for a single kind of free particles, and then find the solution of thisequation in the form of the Kraus representation of the evolution of the density operatorof the system. The next section is devoted to the Heisenberg picture of the evolution of2he same system. In Sect. IV, we analyse the system of particles of different types and theflavour oscillation phenomenon. II. SCHR ¨ODINGER PICTURE
In [35] it was shown that the time evolution of a free unstable scalar particle can be describedby a master equation in the Lindblad–Gorini–Kossakowski–Sudarshan form [26, 27]: dρ ( t ) dt = − i [ H, ρ ( t )] + { K, ρ ( t ) } + Lρ ( t ) L † , (1)where H = m | i h | , L = √ Γ | i h | , K = − L † L . (2)Here | i denotes the state of presence of the particle and | i denotes the state of its absence; m is the mass of the particle and Γ is its decay width. Despite the fact that the state | i is usually called vacuum, it is not the vacuum in the sense of quantum field theory, butrather in the sense used in [39], i.e., it is an absence of a particle. This equation leadsto the probability density of finding the particle evolving according to the Geiger–Nutallexponential law.However, the most natural quantum-mechanical description of systems with variable num-ber of particles is the second quantization formalism. For systems governed by (1) the tran-sition to second quantization is straightforward, since the operators (2) can be interpretedas the vacuum–one-particle sector of the second quantized operatorsˆ H = m ˆ a † ˆ a ≡ m ˆ N , ˆ L = √ Γˆ a , (3)where ˆ a and ˆ a † are bosonic annihilation and creation operators, respectively:[ˆ a, ˆ a † ] = 1 , ˆ a | i = 0 , | n i = (ˆ a † ) n √ n ! | i ; (4)vectors | n i form the so-called occupation number basis. With this substitution we haveˆ K = −
12 Γ ˆ
N .
If we substitute (3) into (1), we arrive to the master equation in the form ddt Λ ∗ t ρ = L ∗ (Λ ∗ t ρ ) , (5a)where L ∗ ( ρ ) = − im [ ˆ N , ρ ] − Γ2 { ˆ N , ρ } + Γˆ aρ ˆ a † . (5b)In the above we can recognize the equation introduced and studied in [29, 31, 32, 34]. Similarequations lead to evolution given by quasi-free semigroups, see eg. [34, 40–47] and are alsostudied in the context of quantum optics [48–51].3f we write down explicitly annihilation and creation operators in occupation numberbasis, namely ˆ a kl = √ k + 1 δ k +1 ,l , ˆ a † kl = √ l + 1 δ k,l +1 , so ˆ N kl = kδ kl ( k, l = 0 , , . . . ), then wecan view (5) as the following infinite system of equations for matrix elements of the densityoperator Λ ∗ t ρ ≡ P kl ρ kl ( t ) | k i h l | : dρ kl ( t ) dt = h i ( k − l ) m − ( k + l )Γ i ρ kl ( t ) + q ( k + 1)( l + 1)Γ ρ k +1 ,l +1 ( t ) , (6)for k, l = 0 , , . . . . Thus we get infinite, in principle, system of linear differential equationsof first order.Notice, that the system (6) seems to be highly non-trivial — the solution for ρ kl ( t ) dependson a solution for ρ k +1 ,l +1 ( t ), what apparently leads to infinite chain of dependencies. Whatmakes the system (6) solvable is the proper choice of initial conditions. Indeed, for everyreasonable initial physical state, the number of particles must be finite, so all matrix elementsof ρ (0) corresponding to higher number of particles must vanish. Mathematically, it meansthat there exist indices r and s , such that ρ kl (0) = 0 for k > r and l > s . (7)One can easily check that the system (6) with initial condition (7) gives us the well posedCauchy’s problem.Now, instead directly solving the equation (5) for some interesting choices of initial statewe concentrate on finding and studying the Kraus representation [52] of the evolution of thesystem.Although Kraus operators for the evolution of the density operator governed by themaster equation (5) was found in [53], here we give its another formulation, now written interms of annihilation/creation operators. It can be easily checked that these two choices ofKraus operators coincide up to the phase factors. Despite this, we give the formal proof thatproposed Kraus operators lead to the evolution of the system undergoing the equation (5),since we will employ the technique used in the proof later on. Proposition 1.
If for k = 0 , , . . . : E k ( t ) = 1 √ k ! e − i ˆ Mt (cid:18)q − e − Γ t ˆ a (cid:19) k , (8) where ˆ M = ( m − i Γ) ˆ
N , then Λ ∗ t ρ = ∞ X k =0 E k ( t ) ρE † k ( t ) (9) is the solution of the master equation (5) , where ρ is the density operator given at initialtime t = 0 .Proof. First, let us note that for any kE k ( t )ˆ a = e ( im + Γ) t ˆ aE k ( t ) , (10)what follows immediately from canonical commutation relations. Using (10) we can showby straightforward calculation that dE ( t ) dt = − i ˆ M E ( t ) , (11a) dE k ( t ) dt = − i ˆ M E k ( t ) + √ k Γ e ( im + Γ) t √ − e − Γ t ˆ aE k − ( t ) (11b)4for k = 1 , , . . . ).Next, one can easily check that the following recurrence relations hold for k = 1 , , . . .E k ( t ) = √ − e − Γ t √ k e ( im + Γ) t ˆ aE k − ( t ) . (12)Combining (12) and (11b) we can write (11) in the form dE k ( t ) dt = − i ˆ M + k e − Γ t − e − Γ t ! E k ( t ) , k = 0 , , . . . (13)Now, let us compute the time derivative of the density operator Λ ∗ t ρ given by (9): ddt Λ ∗ t ρ = ∞ X k =0 dE k ( t ) dt ρE † k ( t ) + E k ( t ) ρ dE † k ( t ) dt ! = − i h ˆ M (Λ ∗ t ρ ) − (Λ ∗ t ρ ) ˆ M † i + Γ e − Γ t − e − Γ t ∞ X k =1 kE k ( t ) ρE † k ( t ) . (14)Taking into account (12) the last term in (14) can be written asΓ e − Γ t − e − Γ t ∞ X k =1 kE k ( t ) ρE † k ( t ) = Γˆ a ∞ X k =1 E k − ( t ) ρE † k − ( t )ˆ a † = Γˆ a (Λ ∗ t ρ )ˆ a † . (15)Thus, the density operator obeys the master equation (5).To complete the proof, we have to show that at the time t = 0 the density operator Λ ∗ t ρ given by (9) is ρ . This is trivial point, because obviously E (0) = id and E k (0) = 0 for k = 1 , , . . . It is easy to see that after writing out annihilation operators in occupation number basisthe Kraus operators (8) differ only by phase factors from those found in [53, 54]. Thesephase factors become important when you try to study a flavour oscillation phenomenon(see Sect. IV).
Proposition 2. If E k ( t ) are given by (8) , then ∞ X k =0 E † k ( t ) E k ( t ) = id . (16) Proof.
We start with the observation that for any element of occupation number basis | n i and any non-negative integer k ˆ a k | n i = s n !( n − k )! | n − k i , n ≥ k , , n < k , (17a)and (ˆ a † ) k | n i = s ( n + k )! n ! | n + k i . (17b)5ince E † k ( t ) E k ( t ) = 1 k ! (cid:16) − e − Γ t (cid:17) k (ˆ a † ) k e − Γ ˆ Nt ˆ a k , then, from (17), for any element of the occupation number basis E † k ( t ) E k ( t ) | n i = nk ! (cid:16) − e − Γ t (cid:17) k e − ( n − k )Γ t | n i , (18)when n ≥ k , and E † k ( t ) E k ( t ) | n i = 0, when n < k . Thus, ∞ X k =0 E † k ( t ) E k ( t ) | n i = n X k =0 E † k ( t ) E k ( t ) | n i = n X k =0 nk ! (cid:16) − e − Γ t (cid:17) k e − ( n − k )Γ t | n i = | n i . (19)Since P ∞ k =0 E † k ( t ) E k ( t ) acts as identity on any element of the basis, it must be the identityoperator. Proposition 3.
Vacuum state | i h | is stable under the evolution given by (8) and (9) .Moreover, lim t →∞ Λ ∗ t ρ = | i h | for any density operator ρ .Proof. Indeed, E ( t ) | i = | i and E k ( t ) | i = 0 for k = 1 , , . . . , so the density operator | i h | is stable during the time evolution.For the proof of the second statement, one can find that E k ( t ) | n i = vuut nk ! e − ( im + Γ)( n − k ) t (cid:18)q − e − Γ t (cid:19) k | n − k i , (20)when n ≥ k , and vanishes otherwise. Next, observe that for Γ > t →∞ e − Γ( n − k ) t (cid:18)q − e − Γ t (cid:19) k = , n > k , , n = k , ∞ , n < k , (21)so, lim t →∞ E k ( t ) | n i = , n = k , | i , n = k , (22)and, consequently, lim t →∞ E k ( t ) | n i h n ′ | E † k ( t ) = , n = n ′ ,δ nk | i h | , n = n ′ . (23)Thus, for any density operator ρ , lim t →∞ Λ ∗ t ρ = tr( ρ ) | i h | = | i h | .Explicit solutions of (1) with operators (3) can be deduced from the relationΛ ∗ t | n i h n ′ | = min { n,n ′ } X k =0 vuut nk ! n ′ k ! e − im ( n − n ′ ) t × e − Γ( n + n ′ − k ) t (cid:16) − e − Γ t (cid:17) k | n − k i h n ′ − k | . (24)6f we impose the superselection rule which forbids the superpositions of states with differentnumber of particles, then the density operator for a system consisting of at most n particlesis of the form Λ ∗ t ρ = n X k =0 p k ( t ) | k i h k | , n X k =0 p k ( t ) = 1 . Thus, it is enough to solve the equation (1) for an initial state of the form ρ = | n i h n | for n being some non-negative integer (arbitrary, but finite), because any density operator forthe initial state is a linear combination of such states.If the system is initially in the n -particle pure state, ρ = | n i h n | , then the solution of theequation (1) is Λ ∗ t ρ = n X k =0 nk ! e − ( n − k )Γ t (cid:16) − e − Γ t (cid:17) k | n − k i h n − k | . The average number of particles changes in time as h N ( t ) i = tr h (Λ ∗ t ρ ) ˆ N i = ne − Γ t . We have thus recovered the Geiger–Nutall exponential decay law. It is worth noting thatthe probability that at a time t one finds exactly k particles from initally n ones, has abinomial distribution B ( n, e − Γ t ) with probability e − Γ t of finding a single particle, as it canbe expected. III. HEISENBERG PICTURE
In Heisenberg picture, master equation for the evolution of an observable Ω is of the form ddt Λ t Ω = L (Λ t Ω) , (25a)where L (Ω) = i [ H, Ω] + 12 nh L † , Ω i L + L † [Ω , L ] o . (25b)Note that 12 { [ L † , Ω] L + L † [Ω , L ] } = { K, Ω } + L † Ω L , but the form used in (25b) is usually more convenient when performing calculations in theHeisenberg picture involving creation and annihilation operators.Having a family of Kraus operators (8), the evolution of observable Ω can be written asthe series Λ t Ω = ∞ X k =0 E † k ( t )Ω E k ( t ) . (26)This representation is especially useful if we can find the decomposition of the observableinto its matrix elements in occupation basis:Ω = X n,n ′ ω n,n ′ | n i h n ′ | . (27)7 t | n i h n ′ | = ∞ X k =0 vuut n + kn ! n ′ + kn ′ ! e im ( n − n ′ ) t × e − Γ( n + n ′ ) t (cid:16) − e − Γ t (cid:17) k | n + k i h n ′ + k | . (28)Using projectors onto n -particle states, ˆΠ n ≡ | n i h n | , the last equation can be rewritten ina more convenient form Λ t ˆΠ n = 1( e Γ t − n ∞ X k = n kn ! (cid:16) − e − Γ t (cid:17) k ˆΠ k . (29) Proposition 4. lim t →∞ Λ t ˆΠ = id .Proof. From (29) it follows thatΛ t ˆΠ = ∞ X k =0 (cid:16) − e − Γ t (cid:17) k ˆΠ k , so lim t →∞ Λ t ˆΠ = ∞ X k =0 ˆΠ k ≡ id.Physically, Proposition 4 tells us that after substantially long (mathematically infinite) pe-riod of time, the probability of finding vacuum reaches one, irrespectively of the state of thesystem. In other words, at infinite time all the Fock spaces collapse to the vacuum subspace.The evolution of creation and annihilation operators can be easily find with use of rela-tion (10): Λ t ˆ a = e − ( im + Γ) t ˆ a , (30a)Λ t ˆ a † = e ( im − Γ) t ˆ a † . (30b)Moreover, it is easy to check that in this case Λ t ˆ N = Λ t ˆ a † Λ t ˆ a (what, in general, does nothold). Indeed, using (10) we haveΛ t ˆ N = ∞ X k =0 E † k ( t )ˆ a † ˆ aE k ( t ) = ∞ X k =0 E † k ( t )ˆ a † E k ( t ) ! e − i ( m + Γ) t ˆ a = Λ t ˆ a † Λ t ˆ a . (31)Consequently, the evolution of the particle number observable isΛ t ˆ N = e − Γ t ˆ N ; (32)we can get this result by solving (25a) for ˆ N , too.It is easy to find the mean number of particles for a given state with help of (32). Here,we consider two examples: the pure state of exactly n particles and a coherent state withgiven mean number of particles ¯ n . Example . If the system is in the pure state of n particles, then the mean number of particlesis simply h N ( t ) i = ne − Γ t . (33)8hus we get the exponential decay law again. Time evolution of the probability of findingexactly k particles follows from (29) and reads p n ( k, t ) = nk ! e − k Γ t (cid:16) − e − Γ t (cid:17) n − k , (34)i.e., it is given by the binomial distribution B ( n, e − Γ t ). Example . Let us assume that the system is in a coherent state | α i , a | α i = α | α i ,α ∈ C , i.e. | α i = e − | α | ∞ X k =0 α k √ k ! | k i , (35)then h N ( t ) i = ¯ ne − Γ t , (36)where ¯ n ≡ | α | is the mean number of particles in the coherent state | α i . Probability offinding exactly k particles evolves in time according to p ¯ n ( k, t ) = 1 k ! (cid:16) ¯ ne − Γ t (cid:17) k e − ¯ ne − Γ t , (37)which is the Poisson distribution P (¯ ne − Γ t ).Let us note that, if we consider the state being a mixture of k -particle states with prob-ability that k -particle state occurs given by the Poisson distribution with mean number ofparticles ¯ n , i.e., ρ = ∞ X k =0 e − ¯ n ¯ n k k ! | k i h k | , (38)then the mean number of particles in this state and probability of finding exactly k particlesare given by the formulae (36) and (37), respectively (despite the fact, that in this case wemust find the traces of the product of observables with the density operator). IV. PARTICLES OF DIFFERENT TYPES
Let us consider a system of particles of r different types (or carrying a quantum numberwith r possible values), each type with mass m j and width Γ j for j = 1 , . . . , r . For such asystem we have [ˆ a j , ˆ a k ] ∓ = 0 , [ˆ a j , ˆ a † k ] ∓ = δ jk , (39)for j, k = 1 , . . . , r , where [ · , · ] ∓ denotes commutator/anti-commutator, respectively, and anti-commutators apply only if both j th - and k th -particles are fermions. The states spanning theoccupation number representation are generated from the vacuum state via the formula | n , n , . . . , n r i = (ˆ a † ) n (ˆ a † ) n · · · (ˆ a † r ) n r √ n ! n ! · · · n r ! | i (40)where we identify | , , . . . , i ≡ | i . 9he master equation for the system takes the following forms ddt Λ ∗ t ρ = − i [ ˆ H, Λ ∗ t ρ ] + { ˆ K, Λ ∗ t ρ } + r X j =1 ˆ L j (Λ ∗ t ρ ) ˆ L † j , (41a) ddt Λ t Ω = i [ ˆ H, Λ t Ω] + r X j =1 n [ ˆ L † j , Λ t Ω] ˆ L j + ˆ L † j [Λ t Ω , ˆ L j ] o , (41b)in the Schr¨odinger and Heisenberg picture, respectively, where ˆ H is the Hamiltonian of thesystem and ˆ L j = q Γ j ˆ a j , ˆ K = − r X j =1 ˆ L † j L j , ˆ M = ˆ H + i ˆ K . (42)If [ ˆ
M , ˆ a j ] = − ( m j − i Γ j )ˆ a j for j = 1 , . . . , r , then we can easily construct the Kraus operatorssolving (41) E k ( t ) = e − i ˆ Mt Y k ,...,k r k + ··· + k r = k (cid:16) √ − e − Γ j t ˆ a j (cid:17) k j q k j ! , (43)where the product is taken over all possible partitions of k into exactly r addends, such that k + k + · · · + k r = k , where k j ∈ N , j th -particles are bosons , (44a) k j ∈ { , } , j th -particle is a fermion . (44b)for j = 1 , . . . , r . Example . Let us consider the evolution of a system of two flavour particles (e.g., particlesand their anti-particles). We denote the creation operators for these particles by ˆ a † and ˆ a † .The basis for the system is built up from the states of the form | n , n i = (ˆ a † ) n (ˆ a † ) n √ n ! n ! | i . (45)Let these states be the common eigenstates of two observables, ˆ N = ˆ a † ˆ a + ˆ a † ˆ a (number ofparticles) and ˆ S = ˆ a † ˆ a − ˆ a † ˆ a (say strangeness or lepton number), i.e.,ˆ N | n , n i = ( n + n ) | n , n i , ˆ S | n , n i = ( n − n ) | n , n i . If the states (45) are not eigenstates of the time evolution the phenomenon known as theflavour oscillation may occur.To describe such a situation, let us assume that the Hamiltonian and Lindblad operatorsfor the system are of the form ˆ H = m ˆ c † ˆ c + m ˆ c † ˆ c , (46a)ˆ L = q Γ ˆ c , (46b)ˆ L = q Γ ˆ c , (46c)10here ˆ c † , ˆ c † are connected with ˆ a † , ˆ a † by unitary transformation:ˆ c † = e iχ e i ( φ + ψ ) / cos θ a † + e − i ( φ − ψ ) / sin θ a † ! , (47a)ˆ c † = e iχ − e i ( φ − ψ ) / sin θ a † + e − i ( φ + ψ ) / cos θ a † ! . (47b)Since ˆ M = (cid:16) m − i Γ (cid:17) ˆ c † ˆ c + (cid:16) m − i Γ (cid:17) ˆ c † ˆ c , we can easily find the evolution of ˆ c j :Λ t ˆ c j = e − ( im j + Γ j ) t ˆ c j , j = 1 , . (48)Using (47) we get the evolution of ˆ a j :Λ t ˆ a = 12 e − ( im + Γ ) t h ˆ a (1 + cos θ ) + ˆ a e iφ sin θ i + 12 e − ( im + Γ ) t h ˆ a (1 − cos θ ) − ˆ a e iφ sin θ i , (49a)Λ t ˆ a = 12 e − ( im + Γ ) t h ˆ a (1 + cos θ ) − ˆ a e − iφ sin θ i + 12 e − ( im + Γ ) t h ˆ a (1 − cos θ ) + ˆ a e − iφ sin θ i (49b)The time evolution of the observables can be obtained either by solving (41b) or directlyfrom relations (49), using argumentation analogous to (31). For example, for the number ofparticles we get Λ t ˆ N = e − Γ t + e − Γ t N + e − Γ t − e − Γ t S cos θ + ˆ Q + sin θ ] , (50)where ˆ Q + = ˆ a † ˆ a e iφ + ˆ a † ˆ a e − iφ , so the mean value in the state | n , n i is the following h N ( t ) i = e − Γ t + e − Γ t n + n ) + e − Γ t − e − Γ t n − n ) cos θ (51)and is depicted in the Fig. 1.Similarly, for the strangeness (or lepton number) we getΛ t ˆ S = e − Γ t − e − Γ t N cos θ + e − Γ t sin(∆ mt ) ˆ Q − sin θ + " e − Γ t + e − Γ t θ + e − Γ t cos(∆ mt ) sin θ ˆ S + " e − Γ t + e − Γ t − e − Γ t cos(∆ mt ) ˆ Q + sin θ cos θ , (52)where ˆ Q − = i (ˆ a † ˆ a e iφ − ˆ a † ˆ a e − iφ ), Γ = (Γ + Γ ) and ∆ m = m − m . The mean value ofthis observable in the state | n , n i is h S ( t ) i = e − Γ t − e − Γ t n + n ) cos θ + " e − Γ t + e − Γ t θ + e − Γ t cos(∆ mt ) sin θ ( n − n ) (53)11 ‡ Θ ‡ Π t Z N ` H t L ^ FIG. 1. Number of particles for system with two flavours for mixing angles θ = 0 , π , π , π , π (fromright to left) with n = 2 and n = 1, and Γ < Γ (time unit is τ = 1 / Γ). and is shown in the Fig. 2.Let us pay our attention on the two extreme cases: θ = 0 (no flavour mixing) and θ = π (maximal mixing). For θ = 0 we have ˆ c i = ˆ a i , so the time evolution of the observables isΛ t ˆ N = 12 e − Γ t ( ˆ N + ˆ S ) + 12 e − Γ t ( ˆ N − ˆ S ) , (54a)Λ t ˆ S = 12 e − Γ t ( ˆ S + ˆ N ) + 12 e − Γ t ( ˆ S − ˆ N ) . (54b)Their mean values in the state | n , n i are h N ( t ) i = e − Γ t n + e − Γ t n , (55a) h S ( t ) i = e − Γ t n − e − Γ t n . (55b)For θ = π , φ = 2 π , ψ = π and χ = π we haveˆ c † = 1 √ a † + ˆ a † ) , ˆ c † = 1 √ a † − ˆ a † ) , and the time evolution of the observables is given byΛ t ˆ N = 12 (cid:16) e − Γ t + e − Γ t (cid:17) ˆ N + 12 (cid:16) e − Γ t − e − Γ t (cid:17) ˆ Q + , (56a)Λ t ˆ S = e − Γ t cos(∆ mt ) ˆ S + e − Γ t sin(∆ mt ) ˆ Q − . (56b)12 ‡ Θ ‡ Π (cid:144) Θ ‡ Π (cid:144) Θ ‡ Π (cid:144) Θ ‡ Π - t Z S ` H t L ^ FIG. 2. Strangeness of system with two flavours for different values of mixing angle with n = 2and n = 1, and Γ < Γ (time unit is τ = 1 / Γ).
The mean values of these observables in the state | n , n i are h N ( t ) i = 12 (cid:16) e − Γ t + e − Γ t (cid:17) ( n + n ) , (57a) h S ( t ) i = e − Γ t cos(∆ mt )( n − n ) , (57b)so we get oscillations of the quantum number S . It is worth noting, that for the particlessuch as K or B mesons we can use this result only as a first approximation, since for theseparticles the transformation which “diagonalizes” the master equation is non-unitary due to CP -violation. For the sake of brevity, we do not discuss the violated CP -symmetry herebut preliminary calculations show the agreement with values for masses and life-times ofneutral K or B mesons estimated on the basis of traditional Wigner–Weisskopf approach. V. CONCLUSIONS
We have analyzed a class of master equations built up from creation and annihilation op-erators which generate dynamical semigroups that can describe the exponential decay andflavour oscillations for system of many particles. We have shown, in this case, how thisdynamical semigroup can be written in the Schr¨odinger as well as Heisenberg picture. Thisallowed us to choose the picture which seems to be more convenient for the description ofthe system under consideration. Moreover, we have found the solution for a free particlemaster equation in the form of Kraus representation in the language of annihilation and13reation operators. Although, this Kraus representation is given by an infinite series, in theSchr¨odinger picture it reduces to a finite sum, whenever the initial state has a finite numberof particles. On the other hand, in the Heisenberg picture the commutation relations be-tween observables and Kraus operators sometimes allows us to find the observable evolutionin closed form without explicit summation of the series.Notice that if we cut the presented approach to the one-zero particle sector we get thetheory given in [35] (neglecting the decoherence).In the present paper we restrict our analysis only to states labeled by a discrete index,and not by continuous parameter (like e.g. momentum). Despite the fact that introducing acontinuous parameter causes creation and annihilation operators to become operator-valueddistributions, it seems to us that the approach introduced here should also be applicable.We left open the question whether it is possible to apply our approach to describe theprocesses other than exponential decay, like e.g. decoherence or different decay laws. Thepreliminary investigations suggest that there exists a positive answer.
ACKNOWLEDGMENTS
This work was supported by the Polish Ministry of Science and Higher Education underContract No. NN202 103738. [1] V. Weisskopf and E. Wigner, Z. Phys. , 54 (1930).[2] V. Weisskopf and E. Wigner, Z. Phys. , 18 (1930).[3] T. Durt, Int. J. Mod. Phys. B , 1345015 (2013).[4] F. Uchiyama, Phys. Lett. A , 295 (1997).[5] A. Bramon and M. Nowakowski, Phys. Rev. Lett. , 1 (1999).[6] R. Foadi and F. Selleri, Phys. Rev. A , 012106 (1999).[7] R. H. Dalitz and G. Garbarino, Nucl. Phys. B , 483 (2001).[8] R. Bertlmann, W. Grimus, and B. C. Hiesmayr, Phys. Lett. A , 21 (2001).[9] R. A. Bertlmann and B. C. Hiesmayr, Phys. Rev. A , 062112 (2001).[10] A. Bramon and G. Garbarino, Phys. Rev. Lett. , 160401 (2002).[11] R. A. Bertlmann, K. Durstberger, and B. C. Hiesmayr, Phys. Rev. A , 012111 (2003).[12] M. Genovese, Phys. Rev. A , 022103 (2004).[13] A. Go, J. Mod. Opt. , 991 (2004).[14] A. Bramon, R. Escribano, and G. Garbarino, J. Mod. Opt. , 1681 (2005).[15] A. Go et al. , Phys. Rev. Lett. , 131802 (2007).[16] F. Benatti and R. Floreanini, Phys. Lett. B , 100 (1996).[17] F. Benatti and R. Floreanini, Nucl. Phys. B , 335 (1997).[18] F. Benatti and R. Floreanini, Phys. Lett. B , 337 (1997).[19] F. Benatti and R. Floreanini, Nucl. Phys. B , 550 (1998).[20] F. Benatti and R. Floreanini, Phys. Rev. D , R1332 (1998).[21] F. Benatti, in Irreversibility and Causality Semigroups and Rigged Hilbert Spaces , LectureNotes in Physics, Vol. 504, edited by A. Bohm, H.-D. Doebner, and P. Kielanowski (Springer,Berlin–Heidelberg, 1998) pp. 124–147.[22] F. Benatti, R. Floreanini, and R. Romano, Nucl. Phys. B , 541 (2001).
23] J. Ellis, J. L. Lopez, N. E. Mavromatos, and D. V. Nanopoulos, Phys. Rev. D , 3846 (1996).[24] J. Bernab´eu, N. E. Mavromatos, and P. Villanueva-P´erez, Phys. Lett. B , 269 (2013).[25] A. Kossakowski, Rep. Math. Phys. , 247 (1972).[26] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, J. Math. Phys. , 821 (1976).[27] G. Lindblad, Commun. Math. Phys. , 119 (1976).[28] A. Kossakowski, in Irreversibility and Causality Semigroups and Rigged Hilbert Spaces , LectureNotes in Physics, Vol. 504, edited by A. Bohm, H.-D. Doebner, and P. Kielanowski (Springer,Berlin–Heidelberg, 1998) pp. 59–66.[29] R. Alicki and M. Fannes,
Quantum Dynamical Systems (Oxford University Press, 2001).[30] H.-P. Breuer and F. Petruccione,
The Theory of Open Quantum Systems (Oxford UniversityPress, 2002).[31] R. Alicki, Rep. Math. Phys. , 1 (1977).[32] R. Alicki, Rep. Math. Phys. , 27 (1978).[33] A. Weron, A. K. Rajagopal, and K. Weron, Phys. Rev. A , 1736 (1985).[34] R. Alicki, in Quantum Dynamical Semigroups and Applications , Lecture Notes in Physics, Vol.717, edited by R. Alicki and K. Lendi (Springer, Berlin–Heidelberg, 2007) pp. 1–46.[35] P. Caban, J. Rembieli´nski, K. A. Smoli´nski, and Z. Walczak, Phys. Rev. A , 032106 (2005).[36] R. A. Bertlmann, W. Grimus, and B. C. Hiesmayr, Phys. Rev. A , 054101 (2006).[37] P. Caban et al. , Phys. Lett. A , 6 (2006).[38] P. Caban, J. Rembieli´nski, K. A. Smoli´nski, and Z. Walczak, Phys. Lett. A , 389 (2007).[39] P. Caban, J. Rembieli´nski, K. A. Smoli´nski, and Z. Walczak, J. Phys. A: Math. Gen. ,3265 (2002).[40] B. Demoen, P. Vanheuverzwijn, and A. Verbeure, Lett. Math. Phys. , 161 (1977).[41] D. E. Evans and J. T. Lewis, J. Funct. Anal. , 369 (1977).[42] P. Vanheuverzwijn, Ann. Inst. Henri Poincar´e (A) , 123 (1978).[43] D. E. Evans, Commun. Math. Phys. , 53 (1979).[44] P. Blanchard, M. Hellmich, P. Lugiewicz, and R. Olkiewicz, J. Math. Phys. , 012106 (2007).[45] P. Blanchard, M. Hellmich, P. Lugiewicz, and R. Olkiewicz, J. Funct. Anal. , 1453 (2009).[46] P. Blanchard, M. Hellmich, P. Lugiewicz, and R. Olkiewicz, J. Funct. Anal. , 2455 (2010).[47] M. Hellmich, Rep. Math. Phys. , 277 (2010).[48] R. Olkiewicz and M. ˙Zaba, Phys. Lett. A , 3176 (2008).[49] R. Olkiewicz and M. ˙Zaba, Phys. Lett. A , 4985 (2008).[50] Y.-H. M. Q.-X. Mu, G.-H. Yang, and L. Zhou, J. Phys. B: At. Mol. Opt. Phys. , 215502(2008).[51] R. Olkiewicz and M. ˙Zaba, J. Phys. B: At. Mol. Opt. Phys. , 205540 (2009).[52] K. Kraus, States, Effects and Operations (Springer, Berlin, 1983).[53] Y.-X. Liu, S¸. K. ¨Ozdemir, A. Miranowicz, and N. Imoto, Phys. Rev. A , 042308 (2004).[54] I. L. Chuang, D. W. Leung, and Y. Yamamoto, Phys. Rev. A , 1114 (1997)., 1114 (1997).