Cent Eur J Phys DOI: 10.2478/s11534-013-0326-x Central European Journal of Physics Coherence-controlled stationary entanglement between two atoms embedded in a bad cavity injected with squeezed vacuum Research Article Xiang–Ping Liao1∗ , Jian–Shu Fang1 , Mao–Fa Fang2 College of Science, Hunan University of Technology, Zhuzhou, Hunan 412007, China Department of Physics and Information Science, Hunan Normal University, Changsha, 410081, Hunan, China Received 07 November 2012; accepted 04 November 2013 Abstract: We investigate the entanglement between two atoms in an overdamped cavity injected with squeezed vacuum when these two atoms are initially prepared in coherent states It is shown that the stationary entanglement exhibits a strong dependence on the initial state of the two atoms when the spontaneous emission rate of each atom is equal to the collective spontaneous emission rate, corresponding to the case where the two atoms are close together It is found that the stationary entanglement of two atoms increases with decreasing effective atomic cooperativity parameter The squeezed vacuum can enhance the entanglement of two atoms when the atoms are initially in coherent states Valuably, this provides us with a feasible way to manipulate and control the entanglement, by changing the relative phases and the amplitudes of the polarized atoms and by varying the effective atomic cooperativity parameter of the system, even though the cavity is a bad one When the spontaneous emission rate of each atom is not equal to the collective spontaneous emission rate, the steady-state entanglement of two atoms always maintains the same value, as the amplitudes of the polarized atoms varies Moreover, the larger the degree of two-photon correlation, the stronger the steady-state entanglement between the atoms PACS (2008): 03.65.Ud; 03.67.Mn; 03.65.Yz Keywords: quantum entanglement • the squeezed vacuum • coherent states • quantum control © Versita sp z o.o Introduction Entanglement is a kind of quantum correlation that has played a central role in quantum information It has been found to be an indispensable resource in various quantum information processes [1–5], but the inevitability of the interaction between the system of interest and its environment may cause decoherence and disentanglement Many authors have shown that the collective interaction with a common thermal environment can cause the entanglement of qubits [6–9] ∗ E-mail: liaoxp1@126.com The study of the controlled entanglement dynamics is of Coherence-controlled stationary entanglement between two atoms embedded in a bad cavity injected with squeezed vacuum considerable importance to the prospects of maintaining quantum information [10–16] Model systems that theoretically exhibit the rebirth of entanglement have been proposed and discussed in several cases [17–19] Clark and Parkins [20] have put forward a scheme to controllably entangle the internal states of two atoms trapped in a high-finesse optical cavity by using quantum-reservoir engineering Duan and Kimble have proposed an efficient scheme to engineer multi-atom entanglement by detecting cavity decay through single-photon detectors for generating multipartite entanglement [21] In Ref [22, 23], it has been shown that white noise may play a positive role in generating the controllable entanglement in some specific conditions Malinovsy and Sola [24] have proposed a method of controlling entanglement in a two-qubit system by changing a relative phase of the pulses In a recent paper, Yu and Eberly [25] have shown that two entangled qubits can become completely disentangled in a finite time under the influence of pure vacuum noise In Ref [26], Ficek and Tanaś have investigated the concept of time-delayed creation of entanglement by the dissipative process of spontaneous emission They have found a threshold effect for the creation of entanglement, whereby the initially unentangled qubits can be entangled after a finite time despite the fact that the coherence between the qubits exists for all times In Ref [27], the author has studied the entanglement and the nonlocality of two qubits interacting with a thermal reservoir The results show that the common thermal resevoir can enhance the entanglement of two qubits when two qubits are initially in coherent states In this paper, we investigate the entanglement between two atoms in an overdamped cavity injected with squeezed vacuum when these two atoms are initially prepared in coherent states It is shown that the stationary entanglement exhibits a strong dependence on the initial state of the two atoms when the spontaneous emission rate of each atom is equal to the collective spontaneous emission rate Conversely, when the spontaneous emission rate of each atom is not equal to the collective spontaneous emission rate, the steady-state entanglement of two atoms always maintain the same value as the amplitudes of the polarized atoms varies Moreover, the larger the degree of two-photon correlation, the stronger the steady-state entanglement between the atoms The master equation We study the dynamics of entanglement between two twolevel atoms embedded in an overdamped cavity injected with a broadband squeezed vacuum We assume that two identical atoms are located in a single-mode cavity The Hamiltonian of this system is H = ωa+ a + ωSz + g(a+ S− + aS+ ), (1) where a and a+ are annihilation and creation operators for the cavity field, and Sz and S± are the collective pseu(j) dospin operators, which are defined as Sz = Σ2j=1 Sz and S± = Σ2j=1 S± The squeezing of all modes seen by the atom is difficult to realize experimentally Instead the cavity environment, where only those modes centred around the privileged cavity mode need be squeezed, provides a much more realistic scenario for experimental investigation The simplest situation to examine is the bad cavity limit Moreover, it is experimentally easy to realize a cavity with the squeezed vacuum input on the one side Thus the broadband squeezed vacuum is injected into the cavity via its lossy mirror (the other mirror is assumed to be perfect) Taking the spontaneous emission into account, the time evolution of the system of atom-field interaction is given by the following master equation [28–30] (j) d ρ = −i[H, ρ] + La ρ + Lc ρ, dt (2) La ρ = γ(2S− ρS+ − S+ S− ρ − ρS+ S− ) + (γ12 − γ)(2S− ρS+ + 2S− ρS+ − S+ S− ρ (1) (2) (2) (1) − S+ S− ρ − ρS+ S− − ρS+ S− , (2) (1) (1) (2) (2) (1) (2) (1) (3) Lc ρ = k(N + 1)(2aρa+ − a+ aρ − ρa+ a) + kN(2a+ ρa − aa+ ρ − ρaa+ ) + kMeiθ (2a+ ρa+ − a+2 ρ − ρa+2 ) + kMe−iθ (2aρa − a2 ρ − ρa2 ), (4) where γ is the spontaneous emission rate of each atom, and γ12 is the collective spontaneous emission rate stemming from the coupling between the atoms through the vacuum field, which is dependent on the separation of the atoms If the atomic separation is much larger than the resonant wavelength, then γ12 ≈ 0; if it is much smaller than the resonant wavelength, then γ12 ≈ γ The parameter k denotes the cavity decay constant The parameter N is the mean photon number of the broadband squeezed vacuum field M measures the strength of two-photon correlations They obey the relation M = η N(N + 1), (0 ≤ η ≤ 1) θ is the phase of the squeezed vacuum We term η the degree of two-photon correlation The injected field is an ideal squeezed vacuum when η = or Xiang–Ping Liao, Jian–Shu Fang, Mao–Fa Fang a nonideal one when η = 1; if η = then this implies no squeezing and our cavity field is then equivalently damped by a chaotic field Here, we are interested in the bad-cavity limit; that is, k g γ, but with C1 = g2 /kγ finite C1 is the effective cooperativity parameter of a single atom familiar from optical bistability To ensure the validity of the broadband squeezing assumption, the bandwidth of squeezing must also be large compared to k In the following, it will be convenient for us to use the basis of the collective states [31–33] B = {|e , |s , |a , |g }, where |e |s = |e1 |e2 , = √ (|e1 |g2 + |g1 |e2 ), |a |g = √ (|e1 |g2 − |g1 |e2 ), = |g1 |g2 (5) The most important property of the collective states is that the symmetric state |s and antisymmetric state |a are maximally entangled states We take the initial coherent state of the two atoms as ψ(0) = cos(α)|e1 g2 + sin(α)eiβ |g1 e2 , which can be generated by controlling the relative phase of the external fields [34] Here β is a relative phase Using the BornMarkoff approximation and tracing over the field state [35– 37], and using the atomic basis as B = {|e , |s , |a , |g }, we can write down the time evolution equations of the density matrix elements for the atoms 4g2 4g2 2g2 (N + 1) − 4γ]ρee + Nρss − η N(N + 1) exp(−iθ)ρeg k k k 2g2 η N(N + 1) exp(iθ)ρge , k 4g2 4g2 4g2 [ (N + 1) + 2(γ + γ12 ]ρee − [ (2N + 1) + 2(γ + γ12 )]ρss + η k k k 4g2 4g2 η N(N + 1) exp(iθ)ρge + Nρgg , k k 2(γ − γ12 )ρee − 2(γ − γ12 )ρaa , ˙ee = [− ρ − ˙ss = ρ + ˙aa = ρ N(N + 1) exp(−iθ)ρeg 2g2 4g2 2g2 η N(N + 1) exp(iθ)ρee + η N(N + 1) exp(iθ)ρss + [− (2N + 1) − 2γ]ρeg k k k 2g η N(N + 1) exp(iθ)ρgg , k 2g2 4g2 2g2 − η N(N + 1) exp(−iθ)ρee + η N(N + 1) exp(−iθ)ρss + [− (2N + 1) − 2γ]ρge k k k 2g η N(N + 1) exp(−iθ)ρgg , k 4g 2g2 [ (N + 1) + 2(γ + γ12 )]ρss + 2(γ − γ12 )ρaa − η N(N + 1) exp(−iθ)ρeg k k 2 2g 4g η N(N + 1) exp(iθ)ρge − Nρgg , k k 2(2N + 1) [−2γ − ]ρas , g 2(2N + 1) [−2γ − ]ρsa , g ˙eg = − ρ − ˙ge = ρ − ˙gg = ρ − ˙as = ρ ˙sa = ρ (6) with the condition ρgg + ρee + ρss + ρaa = It is evident that the other matrix elements retain their initial zero values, and only the set of eight equations (6) can have non-zero solutions when the atoms are initially in the above coherent state It is difficult to obtain analytical solutions to the Eqs (6) We use fourth-order RungeKutta method to solve these equations with the relevant initial conditions In order to calculate the concurrence, Coherence-controlled stationary entanglement between two atoms embedded in a bad cavity injected with squeezed vacuum Entanglement atoms between two 0.8 Concurrence we next transform them into the original basis |e1 ⊗ |e2 , |e1 ⊗|g2 , |g1 ⊗|e2 , |g1 ⊗|g2 In section 3, we will explore quantum entanglement between two two-level atoms embedded in a bad cavity injected with a squeezed vacuum 0.6 0.4 0.2 In order to quantify the degree of entanglement, we choose the Wootters concurrence C [38, 39], defined as 0 10 15 t 20 25 30 (a) λ2 − λ3 − λ4 ), (7) where λ1 , , λ4 are the eigenvalues of the non-Hermition ˜ = ρ(σy ⊗ σy )ρ∗ (σy ⊗ σy ) ρ is the density matrix matrix ρ which represents the quantum state The matrix elements are taken with respect to the ‘standard’ eigenbasis |e1 ⊗ |e2 , |e1 ⊗ |g2 , |g1 ⊗ |e2 , |g1 ⊗ |g2 The concurrence varies from C = for unentangled atoms to C = for the maximally entangled atoms First, we consider the case of γ = γ12 = 0.1, corresponding to the case where the two atoms are close together; i.e., the atomic separation is much smaller than the resonant wavelength In Fig 1, we plot the time evolution of the entanglement between two atoms for γ = γ12 = 0.1, η = 1, gk2 = 10 and θ = π, with (a) N = 0.05, (b)N = 0.5, (c)N = 2, when two atoms are initially in different states with the same phase β = 2π/3 From bottom to top, the lines correspond to α = π/2, α = π/2.08, α = π/2.1363, α = π/2.4813, α = π/3 and α = π/4 This figure shows the dynamical entanglement as α varies and phase remains constant at β = 2π/3 It can be seen that the stationary entanglement of two atoms increases as the amplitude of the polarized atoms cos(α) increases (ie α decreases) It reaches its maximum when α = π/4 We also see from Fig 1(a), (b) and (c) that the stationary entanglement decreases with increasing average photon number N However, these steady entangled states are very robust at high temperature (see Fig 1(c)) Figure displays the time evolution of the entanglement between two atoms for N = 0.001, γ = γ12 = 0.1, η = 0.1, k = and θ = π, for different phases β (from bottom to g2 top, the lines correspond to β = 0, β = π/3, β = π/2, β = 2π/3, β = 5π/6 and β = π) and for given α: (a) α = π/2.2941, (b) α = π/10, (c) α = π/4 This figure displays the dynamical entanglement as the relative phase β varies and α remains unchanged, and depicts how the entanglement of the two atoms depends on the relative phase β It can be seen from this figure that 0.8 Concurrence λ1 − 0.6 0.4 0.2 0 10 15 t 20 25 30 (b) 0.8 Concurrence C = max(0, 0.6 0.4 0.2 0 10 15 t 20 25 30 (c) Figure The entanglement of two atoms versus t for γ = γ12 = 0.1 with (a)N = 0.05, (b)N = 0.5, (c)N = when two qubits are initially in different states with the same phase β = 2π/3 From bottom to top, the lines correspond to α = π/2, α = π/2.08, α = π/2.1363, α = π/2.4813, α = π/3, α = π/4 the stationary entanglement increases with the increasing of the relative phase β This is particularly valuable in that it provides us with a feasible way to manipulate and control the entanglement by changing the relative phases In particular, when β = π, the entanglement of all the states (0 < α < π/4) is larger than their initial Xiang–Ping Liao, Jian–Shu Fang, Mao–Fa Fang to interact with the squeezed vacuum all the time This result can be explained as follows: in an interaction picture with respect to H = ωa+ a + ωSz , after some lengthy algebra and tracing over the field state, from the master equation (2) we find the following master equation for the atoms ρa : Concurrence 0.8 0.6 g2 dρa = − (S˜+ S˜− ρa + ρa S˜+ S˜− − 2S˜− ρa S˜+ ) dt k + L a ρa , (8) 0.4 0.2 0 10 20 30 t 40 50 60 (a) La ρa = γ(2S− ρa S+ − S+ S− ρa − ρa S+ S− ) + (γ12 − γ)(2S− ρa S+ + 2S− ρa S+ − S+ S− ρa (1) − (2) (1) S+ S− ρa − (2) (1) (2) ρa S+ S− (2) − (1) (2) (1) ρa S + S − , (1) (2) (9) Concurrence 0.8 where 0.6 S˜+ = µS+ + ν ∗ S− , S˜− = νS+ + µS− , √ N + 1, µ = √ ν = N exp(iθ) 0.4 0.2 0 10 20 30 t 40 50 (b) Concurrence 0.8 0.6 0.4 0.2 0 10 20 30 t 40 50 60 (c) Figure (10) 60 The entanglement of two atoms versus t for γ = γ12 = 0.1, N = 0.001, for different phases β (from bottom to top, the lines correspond to β = 0, β = π/3, β = π/2, β = 2π/3, β = 5π/6, β = π) and given α:(a)α = π/2.2941, (b)α = π/10, (c)α = π/4 values; this indicates the possibility of obtaining steady entangled states with a larger amount of entanglement originating from entangled states with a smaller amount of entanglement It should be noted that the initial state with α = π/4, β = π always maintains maximal entanglement, regardless of time and temperature, as shown in Fig 2(c) The two atoms in this initial state not seem One can see that the (Bell) state |ψ − = √12 (|10 − |01 )(corresponding to the initial condition α = π/4, β = π) is in fact a “dark" state of the system, i.e S− |ψ − = S+ |ψ − = 0, implying that this state is not influenced by coupling to the squeezed vacuum reservoir (hence the straight line in Fig 2(c)) However, when β = 0, Fig 2(a), (b) and (c) show that entanglement can fall abruptly to zero before entanglement recovers to a stationary state value The time at which the entanglement falls to zero is dependent upon the degree of entanglement of the initial state The bigger the initial degree of entanglement, the later the entanglement vanishes This implies that two-atom entanglement may terminate abruptly in a finite time under the influence of the squeezed vacuum This phenomenon is referred to as “sudden death" of entanglement [25] and it has elucidated a number of new characteristics of entanglement evolution in systems of two qubits Figure displays the entanglement of two atoms versus t for N = 0.05, γ = γ12 = 0.1, η = 1, α = π/10 and θ = π, for different phases β (from bottom to top, the lines correspond to β = 0, β = π/3, β = π/2, β = 2π/3, β = 5π/6, β = π) and given gk2 : (a) gk2 = 4, (b) gk2 = 10, (c) gk2 = 100, (d) gk2 = 1000 This figure displays the dynamical entanglement as gk2 (cf the effective atomic cooperativity parameter C1 = g2 /kγ) varies It is shown that the stationary entanglement of two atoms increases with increasing gk2 1 0.8 0.8 Concurrence Concurrence Coherence-controlled stationary entanglement between two atoms embedded in a bad cavity injected with squeezed vacuum 0.6 0.6 0.4 0.4 0.2 0.2 0 10 20 30 t 40 50 60 10 20 30 t 40 50 60 (b) 1 0.8 0.8 Concurrence Concurrence (a) 0.6 0.6 0.4 0.4 0.2 0.2 0 10 20 30 t 40 50 60 10 20 30 t 40 50 60 (c) Figure (d) The entanglement of two atoms versus t for γ = γ12 = 0.1, N = 0.05, α = π/10, for different phases β (from bottom to top, the lines correspond to β = 0, β = π/3, β = π/2, β = 2π/3, β = 5π/6, β = π) and given k2 : (a) k2 = 4, (b) k2 = 10, (c) k2 = 100,(d) k2 = 1000 g (ie decreasing C1 = g2 /kγ) From figure 3(d), we can see that, when the parameter C1 is very small, which corresponds to the field’s inside cavity being more chaotic, the stationary entanglement of two atoms is very large This is possible, since dissipation plays a crucial role in the generation of the stationary entanglement From figure 3(a), we have found an interesting phenomenon: when β = and gk2 is not large, the entanglement can fall abruptly to zero twice before entanglement recovers to a stationary state value We see two time intervals (dark periods) at which the entanglement vanishes and two time intervals at which the entanglement revives And, with the increase of β, though the phenomenon of “sudden death" of entanglement does not occur, the rate of evolution of entanglement can suddenly change twice Meanwhile, with the increase of gk2 , the entanglement can fall abruptly to zero only once before entanglement recovers to a stationary state value when β = (see Figure 3(b)–(d)) Furthermore, the bigger the parameter gk2 , the shorter the state will stay in the disentangled separable state So, we can steer the evolution of entanglement between two atoms by varying the g g g g effective atomic cooperativity parameter C1 of the system Next, we discuss the situation of γ = γ12 , which means that the separation between two atoms is not very small In Fig 4, we plot the entanglement of two atoms versus t for γ = 0.1, γ12 = 0.06, N = 0.05, gk2 = 10 and θ = π, with (a) η = 1, (b)η = 0.7, (c)η = 0.2 when two atoms are initially in different states with the same phase β = 2π/3 From bottom to top, the lines correspond to α = π/2, α = π/2.08, α = π/2.1363, α = π/2.4813, α = π/3 and α = π/4 When the degree of two-photon correlation η = 1, the injected field in the cavity is an ideal squeezed vacuum, corresponding to the reservoir being in an ideal or minimum uncertainty squeezed state When the degree of two-photon correlation η = 1, the injected field in the cavity is not ideal, which means that some of the photon pairs in the squeezed field are not correlated due to the cavity effect From this figure, which displays the dynamical entanglement as α varies and phase remains constant at β = 2π/3, it is discovered that the steady-state entanglement of two atoms always remains constant as the amplitudes of the polarized atoms α vary This result can Xiang–Ping Liao, Jian–Shu Fang, Mao–Fa Fang to zero, and the symmetric state |s , ρeg and ρge tend to certain values, regardless of the initial states of the atoms In addition, from Fig 3.(a), (b) and (c), we can see that the larger the degree of two-photon correlation η, the stronger the steady-state entanglement between the atoms Thus, the nonclassical two-photon correlations of the injected squeezed vacuum are significant for the stationary entanglement in the system Concurrence 0.8 0.6 0.4 0.2 Conclusion 0 20 40 60 80 100 t (a) Concurrence 0.8 0.6 0.4 0.2 0 20 40 60 80 100 t (b) Concurrence 0.8 0.6 0.4 In this paper, we have investigated the entanglement between two atoms in an overdamped cavity injected with squeezed vacuum when these two atoms are initially prepared in coherent states It is shown that the stationary entanglement exhibits a strong dependence on the initial state of the two atoms when γ = γ12 , corresponding to the case where the two atoms are close together It is found that the stationary entanglement of two atoms increases with decreasing effective atomic cooperativity parameter The squeezed vacuum can enhance the entanglement of two atoms when two atoms are initially in coherent states Valuably, this provides us with a feasible way to manipulate and control the entanglement by changing the relative phases and the amplitudes of the polarized atoms, and by varying the the effective atomic cooperativity parameter of the system even though the cavity is a bad one When γ = γ12 , the steady-state entanglement of two atoms always remains constant as the amplitudes of the polarized atoms α vary Moreover, the larger the degree of two-photon correlation η, the stronger the steady-state entanglement between the atoms Thus, the nonclassical two-photon correlations are significant for the entanglement in the system 0.2 Acknowledgement 0 20 40 60 80 100 t (c) Figure The entanglement of two atoms versus t for γ = 0.1, γ12 = 0.06 with (a)η = 1, (b)η = 0.7, (c)η = 0.2 when two qubits are initially in different states with the same phase β = 2π/3 From bottom to top, the lines correspond to α = π/2, α = π/2.08, α = π/2.1363, α = π/2.4813, α = π/3, α = π/4 be explained as follows: when γ = γ12 , Eqs.(6) imply that ρaa = ρee Therefore, regardless of whether the asymmetric state |a is initially populated, in the long-time limit, due to the interaction of the nonclassical field, the asymmetric state will be equally as populated as the upper lever Given a long time, the values of ρas and ρsa tend This work is supported by the National Natural Science Foundation of China (Grant No 11074072 and No.61174075), Hunan Provincial Natural Science Foundation of china (Grant No 10JJ3088 and No.11JJ2038) and by the Major Program for the Research Foundation of Education Bureau of Hunan Province of China (Grant No 10A026) References [1] P.W Shor, Phys Rev A 52, 2493 (1995) [2] D.P Divincenzo, Science 270, 255 (1995) Coherence-controlled stationary entanglement between two atoms embedded in a bad cavity injected with squeezed vacuum [3] L.K Grover, Phys Rev Lett 79, 325 (1997) [4] J.I Cirac, P Zoller, Nature 404, 579 (2000) [5] M.A Nielsen, I.L Chuang, Quantum Computation and Quantum Information (Cambridge University Press, Cambridge, 2000) [6] D Braun, Phys Rev Lett 89, 277901 (2002) [7] M.S Kim, J Lee, D Ahn, P.L Knight, Phys Rev A 65, 040101(R) (2002) [8] B Kraus, J.I Cirac, Phys Rev Lett 92, 013602 (2004) [9] S Schneider, G.J Milburn, Phys Rev A 65, 042107 (2002) [10] S Maniscalco et al., Phys Rev Lett 100, 090503 (2008) [11] S Das, G S Agarwal, J Phys B 42, 141003 (2009) [12] Y Li, B Luo, H Guo, Hong, Phys Rev A 84, 012316 (2011) [13] D Mundarain, M Orszag, Phys Rev A 79, 052333 (2009) [14] X.Y Yu, J.H Li, EPL 92, 40002 (2010) [15] J Li, K Chalapat, G.S Paraoanu, J Low Temp Phys 153, 294 (2008) [16] Y.J Zhang, Z.X Man, Y.J Xia, J Phys B 42, 095503 (2009) [17] M Yonac, T Yu, J.H Eberly, J Phys B 40, 545 (2007) [18] M Al–Amri, G.X Li, R Tan, M.S Zubairy, Phys Rev A 80, 022314 (2009) [19] L Mazzola, S Maniscalco, J Piilo, K.-A Suominen, B.M Garraway, Phys Rev A 79, 042302 (2009) [20] S.G Clark, A.S Parkins, Phys Rev Lett 90, 047905 (2003) [21] L.M Duan, H.J Kimble, Phys Rev Lett 90, 253601 (2003) [22] M.B Plenio, S.F Huelga, Phys Rev Lett 88, 197901 (2002) [23] J.B Xu, S.B Li, New J Phys 7, 72 (2005) [24] V.S Malinovsky, I.R Sola, Phys Rev Lett 93, 190502 (2004) [25] T Yu, J.H Eberly, Phys Rev Lett 93, 140404 (2004) [26] Z Fieck, R Tanaś, Phys Rev A 77, 054301 (2008) [27] X.P Liao, Phys Lett A 367, 436 (2007) [28] R.R Puri, Mathematical Methods of Quantum Optics (Springer, Berlin, 2001) [29] Z Ficek, R Tanas, Phys Rep 372, 369 (2002) [30] G.X Li, K Allaart, D Lenstra, Phys Rev A 69, 055802 (2004) [31] R.H Dicke, Phys Rev 93, 99 (1954) [32] R.H Lehmberg, Phys Rev A 2, 883 (1970) [33] R.H Lehmberg, Phys Rev A 2, 889 (1970) [34] V.S Malinovsky, I.R Sola, Phys Rev Lett 93, 190502 (2004) [35] J.I Cirac, Phys Rev A 46, 4354 (1992) [36] P Zhou, S Swain, Phys Rev A 54, 2455 (1996) [37] G.S Agarwal, W Lange, H Walther, Phys Rev A 48, 4555 (1993) [38] W.K Wootters, Phys Rev Lett 80, 2245 (1998) [39] S Hill, W.K Wootters, Phys Rev Lett 78, 5022 (1997)