Photon--added squeezed thermal states: statistical properties and its decoherence in a photon-loss channel
aa r X i v : . [ qu a n t - ph ] A ug Photon–added squeezed thermal states: statistical propertiesand its decoherence in a photon-loss channel ∗ Xue-xiang Xu , , Li-yun Hu , † , and Hong-yi Fan Department of Physics, Shanghai Jiao Tong University, Shanghai, 200030, China; College of Physics and Communication Electronics, Jiangxi Normal University, Nanchang, 330022, China.
October 11, 2018
Abstract
Using the normally ordered Gaussian form of displaced-squeezed thermal field characteristicof average photon number ¯ n , we introduce the photon-added squeezed thermo state (PASTS)and investigate its statistical properties, such as Mandel’s Q-parameter, number distribution (asa Legendre polynomial), the Wigner function. We then study its decoherence in a photon-losschannel in term of the negativity of WF by deriving the analytical expression of WF for PASTS.It is found that the WF with single photon-added is always partial negative for the arbitraryvalues of ¯ n and the squeezing parameter r . PACS: 03.65.Yz, 42.50.Dv, 03.67.-a, 03.65.WjKeywords: Open quantum systems; Decoherence; Photon-added squeezed thermal states; photon-loss channel; continuous variable systems; IWOP technique
Nonclassicality of fields has been a topic of great interest in quantum optics and quantum informationprocessing [1]. Experimentally, the traditional quantum states, such as Fock states and coherentstates as well as squeezed states, have been generated but there are some limitations in using themfor various tasks of quantum information process [2]. Alternately, it is possible to generate andmanipulate various nonclassical optical fields by quantum superpositions and subtracting or addingphotons from/to traditional quantum states [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].On the other hand, the single mode displaced, squeezed, mixed Gaussian states have been paidenough attention by both experimentalists and theoreticians. Marian et. al [17, 18] investegated thesuperposition of a squeezed thermal radiation and a coherent one. They examined the squeezingproperties of the field using the distribution functions of the quadratures. For Gaussian squeezedstates of light, a scheme is also presented experimentally to measure its squeezing, purity andentanglement [19, 20]. As is well known, dissipative quantum channels tend to deteriorate thedegree of nonclassicality (i.e., render quantum features unobservable). Thus, it is usually necessaryto investigate the decoherence properties in dissipative channels, such as dynamical behaviors of thepartial negativity of Wigner function (WF) and how long a nonclassical field preserves its partial ∗ Work supported by the National Natural Science Foundation of China (Nos.10775097 and 10874174). † Corresponding author. Tel./fax: +86 7918120370. Email address: [email protected];[email protected]. (L-Y Hu). n and the squeezing parameter r . The work isarranged as follows: In section 2, we introduce the state PASTS and derive its normalized constant,and photon number distribution is discussed in section 3. Section 4 is devoted to calculating theWF. In the last section, we explore the decoherence of PASTS in a photon-loss channel by discussingthe evolution of WF. For a displaced-squeezed thermal field, the density operator is ρ s = D ( β ) S ( r ) ρ c S † ( r ) D † ( β ) , (1)where D ( β ) = exp( βa † − β ∗ a ) , and S ( r ) = exp[i r ( QP + P Q ) / , are the displacement operator andthe squeezing operator [23, 24], respectively, β = ( q + i p ) / √ , Q = a + a † √ , P = a − a † √ i , (cid:2) a, a † (cid:3) = 1 , and ρ c = (1 − e − ~ ωkT ) e − ~ ωa † akT , ( k is the Boltzmann constant, T denoting temperature), is qualified to be a density operator ofthermal (chaotic) field, since tr ρ c = 1. For a coherent state | z i = exp (cid:0) za † − z ∗ a (cid:1) | i [25, 26], due to a | z i = z | z i , matrix elements of any normally ordered operators : ˆ O (cid:0) a † , a (cid:1) : (the symbol : : denotesnormally ordering) in the coherent state is easily obtained, i.e, h z | : ˆ O (cid:0) a † , a (cid:1) : | z ′ i = O ( z ∗ , z ′ ) h z | z ′ i = O ( z ∗ , z ′ ) exp ( − | z | + | z ′ | z ∗ z ′ ) , (2)so in Ref.[27, 28] by using the Weyl ordering invariance under similarity transformations and thetechnique of integration within an ordered product of operators (IWOP) Fan et al have converted ρ s to its normally ordered Gaussian form ρ s = 1 τ τ : exp (cid:26) − ( q − Q ) τ − ( p − P ) τ (cid:27) : , (3)where 2 τ = (2¯ n + 1) e r + 1 , τ = (2¯ n + 1) e − r + 1 , (4)and ¯ n is the average photon number for ρ c , i.e. ¯ n = ( e ~ ω/kT − − [29]. The form in Eq.(3) issimilar to the bivariate normal distribution in statistics, which is useful for us to further derive themarginal distributions of ρ s .Theoretically, the PASTS can be obtained by repeatedly operating the photon creation operator a † on a displacement squeezed thermal state, so its density operator is defined as2 m = N − m a † m ρ s a m , (5)where m is a non-negative integer, N m =tr( a † m ρ s a m ) is the normalization constant. Using Eq.(3)we known immediately the normally ordered Gaussian form of ρ m , i.e., ρ m = N − m τ τ : a † m exp (cid:26) − ( q − Q ) τ − ( p − P ) τ (cid:27) a m : . (6)Next we shall determine the normalization constant N m . Using the completness relation ofcoherent states R d zπ | z i h z | = 1 as well as Eq.(2), we have tr ρ m = tr ( ρ m Z d zπ | z i h z | )= N − m τ τ Z d zπ h z | : a † m exp (cid:26) − ( q − Q ) τ − ( p − P ) τ (cid:27) a m : | z i = N − m τ τ Z d zπ z ∗ m z m exp h − A | z | + B ∗ z + Bz ∗ + Cz + Cz ∗ + D i , (7)where we have set A = 12 τ + 12 τ , B = 1 √ (cid:18) qτ + i pτ (cid:19) ,C = − τ + 14 τ , D = − q τ − p τ . (8)Due to tr ρ m = 1 , thus we know N m = e D τ τ Z d zπ z ∗ m z m exp n − A | z | + B ∗ z + Bz ∗ + Cz + Cz ∗ o = ( − m e D τ τ ∂ m ∂A m Z d zπ exp n − A | z | + B ∗ z + Bz ∗ + Cz + Cz ∗ o . (9)Further using the following integral formula [30] Z d zπ exp n ζ | z | + ξz + ηz ∗ + f z + gz ∗ o = 1 p ζ − f g exp (cid:26) − ζξη + ξ g + η fζ − f g (cid:27) , (10)whose convergent condition is Re( ζ ± f ± g ) < Re (cid:16) ζ − fgζ ± f ± g (cid:17) <
0, Eq.(9) can be rewritten asfollows N m = ( − m e D τ τ ∂ m ∂A m ((cid:0) A − C (cid:1) − / exp " A | B | + CB ∗ + CB A − C , (11)which is the normalization constant of PASTS for photon-added number m . In particular, when β = 0 leading to B = D = 0 , Eq.(11) reduces to the following form, N m = ( − m τ τ ∂ m ∂A m (cid:0) A − C (cid:1) − / . (12)Especially when m = 0 , , N = 1, N = (cid:0) τ + τ (cid:1) and N = (cid:0) τ + 2 τ τ + 3 τ (cid:1) , respectively. 3o see clearly the photon statistical properties of the PASTS, we will examine the Mandel’s Q -parameter defined as Q M = (cid:10) a † a (cid:11) h a † a i − (cid:10) a † a (cid:11) , (13)which measures the deviation of the variance of the photon number distribution of the field stateunder consideration from the Poissonian distribution of the coherent state. If Q M = 0 we say thefield has Poissonian photon statistics while for Q > Q <
0) we say that the field has super-(sub-) Poissonian photon statistics.From Eq.(11) and N m =tr( a † m ρ s a m ), we can easily calculate (cid:10) a † a (cid:11) = N m +1 N m − , and (cid:10) a † a (cid:11) = N m +2 N m − N m +1 N m + 2 , thus we obtain the Q -parameter of the PASTS Q M = N m +2 − N m +1 + 2 N m N m +1 − N m − N m +1 − N m N m . (14)It is well known that the negativity of the Q M -parameter refers to sub-Possonian statistics ofthe state. But a state can be nonclassical even though Q M is positive. This case is true for thepresent state. From Fig.1, one can clearly see that for the cases of m = 0 (Fig.1(a)) , Q M is alwayspositive; while for m = 0 (for instance m = 1) and a given ¯ n value, Q M becomes positive only whenthe squeezing parameter r is more than a certain threshold value that increases as m increases. Inaddition, from Fig.1(a) and Fig.1(b) one can see that the threshold value of r decreases as ¯ n increases.We emphasize that the WF has negative region for all r and ¯ n thus the PASTS is nonclassical (seenext section below). In this section, we study photon number distribution of PASTS optical field. Using the un-normalizedcoherent state | α i = exp[ αa † ] | i , leading to | n i = √ n ! d n d α n | α i | α =0 , (cid:16) h α | α ′ i = e α ′ α ∗ (cid:17) , it is easy tosee that the photon number distribution formula is given by P ( n ) = tr ( ρ m | n i h n | ) = h n | ρ m | n i = 1 τ τ n ! d n d α ∗ n d α ′ n h α | ρ m | α ′ i | α = α ′ =0 . (15)Employing the normal ordering form of ρ m in Eq.(2), Eq.(15) can be put into the following form P ( n ) = N − m e D n ! τ τ d n d α ∗ n d α ′ n (cid:8) α ∗ m α ′ m exp (cid:2) B ∗ α ′ + Bα ∗ + Cα ′ + Cα ∗ + (1 − A ) α ∗ α ′ (cid:3)(cid:9) | α = α ′ =0 . (16)Further expanding the exponential term exp [(1 − A ) α ∗ α ′ ] as series and using the generating functionof single-variable Hermite polynomials, H n ( x ) = ∂ n ∂t n exp (cid:0) xt − t (cid:1) | t =0 , (17)we can calculate the photon number distribution (PND) of PASTS, i.e., P ( n ) = N − m e D n ! τ τ ∞ X l =0 (1 − A ) l l ! ∂ m + l ∂ m + l ∂B m + l ∂B ∗ m + l × d n d α ′ n d α ∗ n exp (cid:8) B ∗ α ′ + Bα ∗ + Cα ′ + Cα ∗ (cid:9) | α = α ′ =0 = N − m | C | n e D n ! τ τ ∞ X l =0 (1 − A ) l l ! (cid:12)(cid:12)(cid:12)(cid:12) ∂ m + l ∂B m + l H n h i B/ (2 √ C ) i(cid:12)(cid:12)(cid:12)(cid:12) . (18)4fter making the scale transformation and noticing the recurrence relation d l d x l H n ( x ) = l n !( n − l )! H n − l ( x ),we can easily obtain P ( n ) = N − m e D τ τ n − m X l =0 n !(1 − A ) l | C | n − m − l l ! [( n − m − l )!] (cid:12)(cid:12)(cid:12) H n − m − l h i B/ (2 √ C ) i(cid:12)(cid:12)(cid:12) , (19)Especially when β = 0, Eq.(19) reduces to P ( n ) = ( − m P [( n − m ) / j =0 n !(1 − A ) ( n − m − j ) | C | j ( n − m − j )!( j !) ∂ m ∂A m ( A − C ) − / = n ! σ n − m P n − m ( − Aσ )( n − m )! ∂ m ∂A m ( A − C ) − / , (20)where σ = q (1 − A ) − C = q ( τ − τ − / ( τ τ ) , (21)and in the last step of (20) we have used the new expression of Legendre polynomials [8] P m ( x ) = x m [ m/ X l =0 m !(1 − x ) l l ( l !) ( m − l )! . (22)Eq. (19) or (20) is just the the analytical expression of the PND of PASTS. In particular, when m = 0 , Eq.(20) becomes (with β = 0) P ( n ) = σ n τ τ P n ( 1 − Aσ ) . (23)Eq.(23) is just the PND of the squeezed thermo state which seems a new result. The PNDs ofPASTS for some given parameters (¯ n, r ) and m are plotted in Fig.2. From Fig. 2 it is found thatthe PND is constrained by n > m . By adding photons, we have been able to move the peak fromzero photons to nonzero photons (see Fig.2 (a)-(c)). The position of peak depends on how manyphotons are created and how much the state is squeezed initially. In addition, comparing Fig.2(b)and Fig.2(d) we see that, for a given m , the “tail” of PND becomes more “wide” with the increasingparameter r . The Wigner function (WF) [31] was first introduced by Wigner in 1932 to calculate quantum cor-rection to a classic distribution function of a quantum-mechanical system. It now becomes a verypopular tool to study the nonclassical properties of quantum states. It is well known that WFsare quasiprobability distributions because it may be negative in phase space [32]. Nevertheless, thepartial negativity of the WF is indeed a good indication of the highly nonclassical character of thestate. Thus, to study the dynamical behaviors of the partial negativity of WF and understand thata nonclassical field preserves its partial negativity, Wigner distribution may be very desirable forexperimentally quantifying the variation of nonclassicality [33].The presence of negativity of the WF for an optical field is a signature of its nonclassicality. Inthis section, using the normally ordered form of PASTS, we evaluate its WF. For a single-modesystem, the WF in the coherent state representation | z i is given by [34]5 ( α, α ∗ ) = e | α | π Z d zπ h− z | ρ m | z i e − zα ∗ − z ∗ α ) , (24)where α = ( x + i y ) / √
2. Then substituting Eq. (6) into Eq. (24) and using Eq. (2), we derive theWF of PASTS W ( α, α ∗ ) = N − m e | α | + D πτ τ ∂ m ∂F m ((cid:0) F − C (cid:1) − / exp " − F | E | + E ∗ C + E CF − C , (25)where we have set F = 2 − A, E = B − α, (26)and used Eq.(10). Especially when β = 0, Eq.(25) reduces to W ( α, α ∗ ) = ( − m e | α | π ∂ m ∂F m n(cid:0) F − C (cid:1) − / exp h − F | α | +4 α ∗ C +4 α CF − C io ∂ m ∂A m ( A − C ) − / , (27)and further when m = 0 , W m =0 ( α, α ∗ ) = 1 π (2¯ n + 1) exp (cid:18) − e − r x + e r y n + 1 (cid:19) , (28)which is the WF of the squeezed thermo state, and W m =1 ( α, α ∗ ) = M x + N y + Υ π exp (cid:18) − e − r x + e r y n + 1 (cid:19) , (29)respectively, where we have set M = 1(2¯ n + 1) (cid:0) τ e − r (cid:1) τ + τ , (30) N = 1(2¯ n + 1) (cid:0) τ e r (cid:1) τ + τ , (31)Υ = 1(2¯ n + 1) τ + τ − τ τ τ + τ . (32)Eq.(28) just agrees with the result of Eq.(48) in Ref. [27], whose form is normal distribution.From Eq.(27) one can see that the WF of the PASTS is always real, as expected. When the factor M x + N y + Υ < m = 1 has its negative distributionin phase space. Noticing M , N always positive, this indicates that the WF of the PASTS alwayshas the negative values under the condition Υ < (cid:0) τ + τ − τ τ < (cid:1) ) at the phase spacecenter q = p = 0. In fact, by substituting Eqs. (4), (30)-(32) into Υ <
0, we find that for thearbitrary values of ¯ n and r , the WF with m = 1 is always partial negative.Using Eq.(27), the WFs of the PASTS are dipicted in phase space for several different values of m, ¯ n and r in phase space. Fig. 3 exhibits the WFs of the PASTS in phase space with m = 1 fordifferent ¯ n , r . It is easy to see that the WFs of single-PASTS always have the negative region. Theminimum in the negative region becomes larger with the increasing of ¯ n (see Fig3.(a) and (c)). InFig. 4, we have presented the WFs with ¯ n = 0 . r = 0 . m, which indicates that thepeak (absolute) value of WF become smaller as the increasing parameter m . The partial negativityof WF indicates the nonclassical nature of the PASTS field.6 Evolution of WF in a photon-loss channel
When the PASTS evolves in the amplitude decay channel, the evolution of the density matrix canbe described by the following master equation in the interaction picture [35], dρdt = κ (cid:0) aρa † − a † aρ − ρa † a (cid:1) , (33)where κ represents the rate of decay. By using the thermal field dynamics theory and thermalentangled state representation, the time evolution of WF at time t to be given by the following form[36], i.e., W ( α, t ) = 2 T Z d zπ e − T | α − ze − κt | W ( z, , (34) T = 1 − e − κt . Eq.(34) is just the evolution formula of WF of single mode quantum state in photon-loss channel. Byobserving Eq.(34), we see that when t → , T → , π T exp( − T | α − ze − κt | ) → δ ( α − z ) δ ( α ∗ − z ∗ ) , so W ( α, t ) → W ( α,
0) as expected. Thus the WF at any time can be obtained by performing theintegration when the initial WF is known.For simplicity, here we only discuss the special case β = 0. Substituting Eq.(27) into Eq.(34) andusing Eq.(10), we derive the time evolution of WF for PASTS in photon-loss channel: W ( α, t ) = 2( − m π T ∂ m ∂F m (cid:16) √ N e R | α | + k α ∗ + k α (cid:17) ∂ m ∂A m ( A − C ) − / , (35)where we have set N = F − C h F + (cid:16) e − κt T − (cid:17) ( F − C ) i − C , R = 4 N e − κt T (cid:20) F + (cid:18) e − κt T − (cid:19) (cid:0) F − C (cid:1)(cid:21) − T , (36) k = 16 N Ce − κt T . In particular, when m = 0 Eq.(35) becomes W m =0 ( α, t ) = 2 p N ( A − C ) π T e R | α | + k α ∗ + k α (37)which is just the WF of the squeezed thermo state in photon-loss channel. This result can also bechecked by substituting Eq.(28) into Eq.(34).When κt exceeds a threshold value, the WF has no chance to be negative in the whole phasespace. At long time κt → ∞ , leading to N → F − − C , R → − , k → , the WF in Eq.(35)becomes W ( α, ∞ ) = e − | α | π , (38)which corresponds to the Gaussian state. In Fig.5, the WFs of PASTS are depicted in phase spacewith m = 1 and r = 0 . κt . It is easily seen that the negative region of WFgradually disappears as κt increases. This implies that the system state reduces to a Gaussian state7fter a long time interaction in the channel. Thus the loss of channel causes the absence of thepartial negativity of the WF if the decay time κt exceeds a threshold value.In Figs. 6, we have also presented the time-evolution of WF for different r . One can see clearlythat the partial negativity of WF decreases gradually as r increases. The squeezing effect in one ofthe quadratures can be seen in Fig.6. In Eq.(35), for the PASTS we have obtained the expressionof the time evolution of WF. In principle, by differentiating as shown in Eq.(35) we can derivethe WF of other PASTS ( m > m = 2 , , m increases. Based on the normally ordered Gaussian form of displaced-squeezed thermal field, we have introduceda kind state: the photon-added squeezed thermo state (PASTS). Then we have investigated thestatistical properties of PASTS (such as Mandel’s Q-parameter, number distribution, the Wignerfunction) and its decoherence in photon-loss channel with dissipative coefficient κ in term of thenegativity of WF by deriving the analytical expression of WF for PASTS. It is found that thephoton number distribution is just a Legendre polynomial and that the WF with single photon-added is always partial negative for the arbitrary values of ¯ n and the squeezing parameter r . Thetechnique of integration within an ordered product of operators brings convenience in our derivation. References [1] D. Bouwmeester, A. Ekert, A. Zeilinger, The Physics of Quantum Information, Springer-Verlag,2000.[2] M. S. Kim, J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 133001-1–18.[3] A. Zavatta, S. Viciani, M. Bellini, Science, 306 (2004) 660-662.[4] A. Zavatta, S. Viciani, M. Bellini, Phys. Rev. A 72 (2005) 023820-1–9.[5] A. Zavatta, V. Parigi, and M. Bellini, Phys. Rev. A 75 (2007) 052106-1–6.[6] A. Biswas and G. S. Agarwal, Phys. Rev. A 75 (2007) 032104-1–8.[7] P. Marek, H. Jeong, M. S. Kim, Phys. Rev. A 78 (2008) 063811-1–8.[8] L. Y. Hu, H. Y. Fan, J. Opt. Soc. Am. B, 25 (2008) 1955-1964.[9] A. Ourjoumtsev, A. Dantan, R. Tualle-Brouri, P. Grangier, Phys. Rev. Lett. 98 (2007) 030502-1–4.[10] J. S. Neergaard-Nielsen, B. Melholt Nielsen, C. Hettich, K. Momer, E. S. Polzik, Phys. Rev.Lett. 97 (2006) 083604-1–4.[11] A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, Ph. Grangier, Science 312 (2006) 83-6.[12] S. Glancy, H. M. de Vasconcelos, J. Opt. Soc. Am. B 25 (2008) 712-733.[13] S. Olivares, Matteo G. A. Paris, J. Opt. B: Quantum Semiclass. Opt. 7 (2005) S392-S397.814] A. Kitagawa, M. Takeoka, M. Sasaki, A. Chefles, Phys. Rev. A 73 (2006) 042310-1–12.[15] L. Y. Hu and H. Y. Fan, Phys. Scr. 79 (2009) 035004-1–8.[16] X. Y. Chen, Phys. Lett. A 372 (2008) 2976-2979 .[17] P. Marian, T. A. Marian, Phys. Rev. A 47 (1993) 4474-4486.[18] P. Marian, T. A. Marian, Phys. Rev. A 47 (1993) 4487-4495.[19] J. Fiurasek, N. J. Cerf, Phys. Rev. Lett. 93 (2004) 063601-1–4.[20] J. Wenger, et. al., Phys. Rev. A 70 (2004) 053812-1–8.[21] S. B. Li, Phys. Lett. A 372 (2008) 6875-6878.[22] L. A. M. Souza, M. C. Nemes, Phys. Lett. A 372 (2008) 3616-3619.[23] M. O. Scully, Zubairy, Quantum optics, Cambridge University Press, 1998.[24] V. V. Dodonov, J. Opt. B: Quantum Semiclass. Opt, 4 (2002) R1-R33.[25] R. J. Glauber, Phys. Rev. 130 (1963) 2529-2539; Phys. Rev. 131 (1963) 2766-2788.[26] J. R. Klauder, B. S. Skargerstam, Coherent States, World Scientific, Singapore, 1985.[27] H. Y. Fan, Annals of Physics, 323 (2008) 1502-1528.[28] H. Y. Fan, T. T. Wang, L. Y. Hu. Chin. Phys. Lett. 25 (2008) 3539-3542.[29] W. H. Louisell, Quantum Statistical Properties of Radiation, Wiley, New York, 1973.[30] H. Y. Fan, From Quantum Mechanics to Quantum Optics-Development of the MathematicalPhysics, Jiao Tong University Press, Shanghai, pp.107, 2005 (in Chinese).[31] E. Wigner, Phys. Rev. 40 (1932) 749-759.[32] W. P. Schleich, Quantum Optics in Phase Space, Wiley-VCH, Berlin, 2001.[33] G. S. Agarwal, E. Wolf, Phys. Rev. D 2 (1970) 2161–2186, 2187–2205, 2206–2225.[34] H. Y. Fan, H. R. Zaidi, Phys. Lett. A 124 (1987) 303-307.[35] C. Garder, P. Zoller, Quantum Noise, Springer, Berlin, 2000.[36] L. Y. Hu, H. Y. Fan, arXiv: quant-ph/0903.2900.Figures caption:Fig.1 Mandel’s Q -parameter of PASTS as a fuction of r with m = 0 , , , ,
30 (from top tobottom) for (a) ¯ n = 0 .
3; (b) ¯ n = 1 . Fig.2 Photon number distributions of PASTS with ¯ n = 1 for (a) r = 0 . , m = 0; (b) r = 0 . , m =1;(c) r = 0 . , m = 5;(d) r = 0 . , m = 1 . Fig. 3 WF of PASTS for m = 1 (a) ¯ n = 0 , r = 0 .
3; (b) ¯ n = 0 , r = 0 . n = 0 . , r = 0 . n = 0 . , r = 0 . . Fig. 4 WF of PASTS for ¯ n = 0 . , r = 0 . m = 0; (b) m = 2;(c) m = 3 ;(d) m = 5 . Fig. 5 The time evolution of WF of PASTS for m = 1 , r = 0 . , and ¯ n = 0 . κt = 0 . κt = 0 . κt = 0 . κt = 0 . . Fig. 6 WF of PASTS for m = 1 , ¯ n = 0 . κt = 0 . , with (a) r = 0 .
01; (b) r = 0 . r = 1 . Fig. 7 WF of PASTS for r = 0 .
3, ¯ n = 0 . κt = 0 . , with (a) m = 2; (b) m = 3 . (c) m = 5 . .1 0.2 0.3 0.4 0.5 r (cid:16) Q M r Q M (b)(a) Figure 1: Mandel’s Q -parameter of PASTS as a fuction of r with m = 0 , , , ,
30 (from top tobottom) for (a) ¯ n = 0 .
3; (b) ¯ n = 1 . P ( n ) n P ( n ) n (a) (b) P ( n ) n P ( n ) n (c) (d) Figure 2: Photon number distributions of PASTS with ¯ n = 1 for (a) r = 0 . , m = 0; (b) r = 0 . , m =1;(c) r = 0 . , m = 5;(d) r = 0 . , m = 1 . (b)(a) (c) (d) Figure 3: WF of PASTS for m = 1 (a) ¯ n = 0 , r = 0 .
3; (b) ¯ n = 0 , r = 0 . n = 0 . , r = 0 . n = 0 . , r = 0 . . (a) (b) (c) (d) Figure 4: WF of PASTS for ¯ n = 0 . , r = 0 . m = 0; (b) m = 2;(c) m = 3 ;(d) m = 5 . (b)(a)(c) (d) Figure 5: The time evolution of WF of PASTS for m = 1 , r = 0 . , and ¯ n = 0 . κt = 0 . κt = 0 . κt = 0 . κt = 0 . . a)(b)(c) Figure 6: WF of PASTS for m = 1 , ¯ n = 0 . κt = 0 . , with (a) r = 0 .
01; (b) r = 0 . r = 1 . a)(b)(c) Figure 7: WF of PASTS for r = 0 .
3, ¯ n = 0 . κt = 0 . , with (a) m = 2; (b) m = 3 . (c) m = 5 ..