En este trabajo proveemos una discusión introductoria sobre cuasi-probabilidades en (óptica cuántica y las usamos para evaluar el parámetro de Mandel.In this work we provide an introductory discussion to quasi-probabilities in quantum optics and how to use them for evaluating the Mandel  parameter.Fil: Plastino, Angel Luis. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico La Plata. Instituto de Física La Plata; Argentina. Universidad Nacional de La Plata; ArgentinaFil: Pennini, A.. Universidad Católica del Norte; Chil

Pennini, A.

Plastino, Angel Luis

Revista Mexicana de Física

In this work we provide an introductory discussion to quasi-probabilities in quantum optics and how to use them for evaluating the Mandel parameter.En este trabajo proveemos una discusion introductoria sobre cuasi-probabilidades en óptica cuántica y las usamos para evaluar el parámetro de Mandel.Instituto de Física La Plat

Plastino, Ángel Luis

Pennini, Flavia

LAReferencia - Red Federada de Repositorios Institucionales de Publicaciones Científicas Latinoamericanas

EDUCATION Revista Mexicana de Fı́sica E 60 (2014) 103–106 JULY–DECEMBER 2014Different creation-destruction operators’ ordering,quasi-probabilities, and Mandel parameterF. Penninia,b, A. PlastinoaaInstituto de Fı́sica La Plata–CCT-CONICET, Universidad Nacional de La Plata,C.C. 727, 1900, La Plata, ArgentinabDepartamento de Fı́sica, Universidad Católica del Norte,Av. Angamos 0610, Antofagasta, Chile.Received 27 August 2014; accepted 27 October 2014In this work we provide an introductory discussion to quasi-probabilities in quantum optics and how to use them for evaluating the Mandelparameter.Keywords: Quantum distributions; phase space; Mandel parameter.En este trabajo proveemos una discusión introductoria sobre cuasi-probabilidades en óptica cuántica y las usamos para evaluar el parámetrode Mandel.Descriptores: Distribuciones cuánticas; espacio de fases; parámetro de Mandel.PACS: 03.67.-a; 05.30.-d; 42.50.-p; 42.50.Lc1. IntroductionThe focus of this article is the notion of quasi-probability.Why? because this is an important concept for quantum op-tics (among other fields). What is a quasi-probability dis-tribution (QPD)? It is a mathematical construction that, al-though resembling a probability distribution, does not exactlyfulfill the Kolmogorov’s axioms on which probability theoryis founded [1], more specifically, the third one may be vio-lated [1]. Quasi-probabilities exhibit general features of or-dinary probabilities. They yield expectation values with re-spect to the weights of the quasi-distribution. However, letus reiterate, they violate the third probability postulate [1], inthe sense that regions integrated under them do NOT rep-resent probabilities of mutually exclusive states. In somecases, quasi-probability distributions exhibit zones of nega-tive probability density. QPDs often arise in the phase spacerepresentation of quantum mechanics, customarily employedin quantum optics, time-frequency analysis, etc.It is well known that the dynamics of a quantum systemis determined by a master equation. One faces an equation ofmotion for the density operator ρ̂, expressed via a completeorthonormal basis. It is always possible to cast the densityoperator in a diagonal manner, provided that an overcompletebasis be used [2]. The most famous such basis is that of co-herent states |α〉 [3]. One writes [2]ρ̂ =∫d2απf(α, α∗) |α〉〈α|. (1)We see that a central role is assigned to the ordinary func-tion f , that becomes endowed with the features of a quasi-probability distribution. In particular, one has d2α/π =dxdp/2π~. The system evolves as prescribed by the evolu-tion of the quasi-probability distribution function. Coherentstates, right eigenstates of the annihilation operator â, serveas the overcomplete basis in such a build-up [2, 3].In quantum optics f is called the function P . One speaksof the P-representation. There exists two other importantrepresentations, known as the Q- and W - ones. There ex-ists a family of different representations, each connected toa different ordering of the underlying creation and destruc-tion operators â and â†. Historically, the first of these isthe Wigner quasi-probability distribution W [4], related tosymmetric operator ordering. In quantum optics the particlenumber operator is naturally expressed in normal order and,in the pertinent scenario, the associated representation of thephase space distribution is the Glauber–Sudarshan P one [3].In addition to these two (W and P ), one often encountersother quasi-probability distributions emerging in alternativerepresentations of the phase space distribution [5]. A quitepopular representation is the Husimi Q one [6-9], employedwhen operators are in anti-normal order.In this paper we wish illustrate how these orderings aretackled so as to find the W , P , and Q representations for theimportant instance of the Harmonic Oscillator (HO) of angu-lar frequency ω in the canonical ensemble formulation. Insuch a scenario one deals with three functions associated tothe W, Q, and P representations that are simple Gaussians andthe treatment becomes entirely analytical, a very convenientdidactic feature. The HO is a really important system thatyields insights usually having a wide impact. Thus, the HOconstitutes much more than a mere simple example. Nowa-days, it is of particular interest for the dynamics of bosonic orfermionic atoms contained in magnetic traps [10-12] as wellas for any system that exhibits an equidistant level spacingin the vicinity of the ground state, like nuclei or Luttingerliquids. For the sake of concreteness we focus attention ona physical important quantity, a noise indicator, called theMandel parameter.So as to accomplish our didactic purposes, this commu-nication is organized as follows. In Sec. 2. we review some104 F. PENNINI AND A. PLASTINOnotions about the Mandel parameter. In Sec. 3. we dis-cuss the formulation of the Mandel parameter in terms ofthe P−function for a thermal state. In Sec. 4. we reproducethe same ideas via Q−function, and in Sec. 5. we recalculatethese results in terms of the Wigner function. Finally, someconclusions are drawn in Sec. 6.2. Mandel parameterSince the very beginnings of quantum mechanics there hasbeen interest in gauging how non-classical a quantum systemcan be. Several questions can be asked in this respect. Forinstance, how closely does the probability distribution for astate after a given series of measurements resemble that of aclassical system? Or, how much is a quantum state perturbedby a measurement on the system?A useful parameter to characterize non-classicality inquantum optics is the so-called Mandel parameter, introducedby Mandel in Ref. 13. It is defined asQM = (∆n̂)2〈n̂〉 − 1 ≡ F − 1, (2)and intimately linked to the normalized variance,which is also denominates the quantum Fano factorF=(∆n̂)2/〈n̂〉 [15] of the photon distribution, with(∆n̂)2 = 〈n̂2〉 − 〈n̂〉2, and the number operator n̂ = â†â.One must take note that• For F < 1 (QM ≤ 0), emitted light is referred to assub-Poissonian since it has photo-count noise smallerthan that of coherent (ideal laser) light with the sameintensity (F = 1; QM = 0), whereas• for F > 1, (QM > 0) the light is called super-Poissonian, exhibiting photo-count noise higher thanthe coherent-light noise.If for the photon number operator n̂ the fluctuations in n̂ dis-appear, the Mandel parameter becomes QM = −1 (F = 0).We pass now to our central issue, the evaluation of the Man-del parameter for our three quasi-probability instances, i.e.,P , Q, and W .3. Mandel Parameter via the P−functionThe most general density operator is just a superpositionof projection operators, known as the Glauber-SudarshanP−representation [14]. One hasρ̂ =∫d2απP (α, α∗) |α〉〈α|, (3)where the function P (α, α∗) plays the role of a probabilitydensity for the distribution of values of α over the complexplane. A coherent state |α〉 is a specific kind of quantum state,the one that most resembles a classical state. It is applicableto the quantum harmonic oscillator, the electromagnetic field,etc., and describes a maximal kind of coherence and a clas-sical kind of behavior. The states |α〉 are normalized, i.e.,〈α|α〉 = 1, and they provide us with a resolution of the iden-tity operator∫d2απ|α〉〈α| = 1, (4)which is a completeness relation for the coherent states [3].The standard coherent states |α〉 for the harmonic oscillatorare eigenstates of the annihilation operator â, with complexeigenvalues α, which satisfy â|α〉 = α|α〉 [3]. Also, â, a†,and n̂ fulfills the commutation relations[â, â†] = 1, (5a)[n̂, â] = −â, (5b)[n̂, â†] = â†. (5c)As stated above, P (α) is a quasi-probability distributionfunction because it can have negative values and strong sin-gularities, especially when the density operator correspondsto a nonclassical state with sub-Poisson photon statistics(see Ref. 16 and references therein). When this functiontends to vary little over large ranges of the parameter α, thenonorthogonality of the coherent states will make little differ-ence, and P (α) can be interpreted as a probability distribu-tion [3]. The normalization property of the density operatorrequires that P (α) obey the normalization condition [3]Tr ρ̂ =∫d2απP (α, α∗) = 1. (6)Accordingly, the expectation value of an observable Â isgiven by [16]〈Â〉 = Tr(ρ̂Â) =∫d2απP (α, α∗) 〈α|Â|α〉. (7)In this context, the average particle-number –an importantquantity for our present considerations– acquires a simpleform that, according to Eq. (3), can be cast in the fashion [3]〈n̂〉 = Tr(ρ̂â†â) = 〈â†â〉P=∫d2απP (α, α∗) |α|2 = 〈|α|2〉P , (8)indicating that the average photon number is the meansquared absolute value of the amplitude α. Note that 〈 〉Pis the average with respect to P (α).In particular, for a thermal state for which the density op-erator is of the form prescribed by the canonical ensemble’srepresentation we haveρ̂ = (1− e−β~ω) e−β~ωâ†â (9)Rev. Mex. Fis. E 60 (2014) 103–106DIFFERENT CREATION-DESTRUCTION OPERATORS’ ORDERING, QUASI-PROBABILITIES, AND MANDEL PARAMETER 105and the corresponding P−function becomesP (α, α∗) =1〈n̂〉 exp(−|α|2〈n̂〉), (10)while the average particle-number is given by [3]〈n̂〉 = 〈|α|2〉P = e−β~ω1− e−β~ω . (11)Now, for all normal-ordered operator-averages we have〈â†râs〉P =∫d2απP (α, α∗)α∗rαs. (12)In particular, for r = s = 2, and taking into account thecommutation relations between â and â†, we find〈â†2â2〉P = 〈(â†â)2〉P − 〈â†â〉P ≡ 〈n̂2〉 − 〈n̂〉, (13)where〈â†2â2〉P =∫d2απP (α, α∗) |α|4 = 〈|α|4〉P . (14)Considering Eqs. (13) and (14), we realize that one can write〈n̂2〉 = 〈|α|2〉P + 〈n̂〉. (15)In accordance with Eqs. (11) and (15), the Mandel pa-rameter now becomes, via the P−function,QM = 〈|α|4〉P − 〈|α|2〉2P〈|α|2〉P =e−β~ω1− e−β~ω , (16)a temperature dependent expression, where the statisticalaverages in phase space are computed utilizing P (α) as aweight function〈|α|s〉P = 2∞∫0dα |α|s+1 P (α, α∗), (17)with s = 2, 4.4. Mandel parameter via the Q−functionA closely related phase space distribution is obtained by tak-ing the diagonal matrix element of density operator ρ̂Q(α, α∗) = 〈α|ρ̂|α〉, (18)which it has all the properties of a classical probability dis-tribution [18]. For a thermal state, the Q−function is thegaussian quantity given byQ(α, α∗) =11 + 〈n̂〉 exp(− |α|21 + 〈n̂〉). (19)The Q−representation gives operator averages in antinor-mal order, so in this case the antinormal-ordered average be-comes [18]〈âsâ†r〉Q =∫d2απQ(α, α∗)α∗rαs, (20)where 〈 〉Q denotes the average with respect to Q(α, α∗).Taking r = s = 2 we have〈â2â†2〉Q =∫d2απQ(α, α∗) |α|4 = 〈|α|4〉Q. (21)Considering the identity â2â†2 = n̂2 + 3n̂ + 2 and Eq. (21),it is easy to show that〈n̂2〉 = 〈|α|4〉Q − 3〈n̂〉 − 2. (22)Also, for s = r = 1, Eq. (20) reduces to〈ââ†〉Q =∫d2απQ(α, α∗) |α|2 = 〈|α|2〉Q, (23)and we have〈n̂〉 = 〈|α|2〉Q − 1. (24)Finally, from Eqs. (22) and (24) the Mandel parameter interms of averages of Q turns out to beQM =〈|α|4〉Q − 2〈|α|2〉Q − 〈|α|2〉2Q + 1〈|α|2〉Q − 1=e−β~ω1− e−β~ω , (25)and coincides with the pertinent expression obtained via theP -representation. Here, the means values are calculated us-ing〈|α|s〉Q = 2∞∫0dα |α|s+1 Q(α, α∗), (26)with s = 2, 4, with Q(α, α∗) given by (19).5. Mandel parameter via the Wigner functionThe Wigner function can be obtained from the P−functionfrom the relation [17]W (α, α∗) = 2∫d2zπP (z, z∗) exp(−2|α− z|2), (27)such that for a thermal state this function becomesW (α, α∗) =1〈n̂〉+ 1/2 exp(− |α|2〈n̂〉+ 1/2), (28)with 1/(〈n̂〉 + 1/2) = 2 tanh(β~ω/2). The symmetric or-dered operator used in this Wigner representation is, in thiscase,〈(â†râs)S〉W =∫d2απW (α, α∗) α∗rαs, (29)Rev. Mex. Fis. E 60 (2014) 103–106106 F. PENNINI AND A. PLASTINOwhere (â†râs)S denotes the symmetric operator product ofthe creation operators and annihilation ones [18], and 〈 〉Windicates the average with respect to W (α, α∗). Thus, forr = s = 2 we get〈(â†2â2)S〉W =∫d2απW (α, α∗) |α|4=16(〈â†2â2〉+〈â†ââ†â〉+ 〈â†â2â†〉+ 〈ââ†2â〉+ 〈ââ†ââ†〉+ 〈â2â†2〉), (30)implying, after to use the commutation relations (5), that〈n̂2〉 = 〈|α|4〉W − 〈n̂〉 − 12 . (31)Also, for r = s = 1 we get,〈(â†â)S〉W =∫d2απW (α, α∗) |α|2=12(〈âa†〉+ 〈â†â〉), (32)yielding〈n̂〉 = 〈|α|2〉W − 12 . (33)From Eqs. (31) and (33) we obtain now the Mandel parame-ter in the fashionQM = 〈|α|4〉W − 〈|α|2〉2W − 〈|α|2〉W + 1/4〈|α|2〉W − 1/2=1− tanh(β~ω/2)2 tanh(β~ω/2)=e−β~ω1− e−β~ω , (34)where we have considered the mean values of |α|2 and |α|4according to〈|α|s〉W = 2∞∫0dα |α|s+1 W (α, α∗), (35)with s = 2, 4, being W (α, α∗) a statistical wight functiongiven by (28).Interestingly enough, the Mandel parameter for a thermalstate is the same in the three representations we are discussingin this work.6. ConclusionsWe have shown into some detail how to proceed to evalu-ate the quasi-probability distributions corresponding, respec-tively, to the P-, Q- and Wigner representations in the case ofa thermal state for the harmonic oscillator. Using them wehave computed the noise-factor called the Mandel parameter.We have seen that it is the same quantity, independent on thenature of the quasi-probability distribution. It is hoped thatthese reflections may help students in their efforts to navigatequantum optics waters.AcknowledgementF. Pennini would like to thank partial financial support byFONDECYT, grant 1110827.This work was also partially supported by the projectPIP1177 of CONICET (Argentina), and the projectsFIS2008-00781/FIS (MICINN)-FEDER (EU) (Spain, EU).1. J. Von Plato, Grundbegriffe der Wahrscheinlichkeitsrechnungin Grattan-Guinness, I., (ed., Landmark Writings in WesternMathematics, Elsevier, Amsterdam, pp 960-69, 2005).2. E.C.G. Sudarshan, Phys. Rev. Lett. 10 (1963) 277.3. R.J. Glauber, Phys. Rev. 131 (1963) 2766.4. E.P. Wigner, Phys. Rev. 40 (1932) 749.5. F. Pennini, A. Plastino, Eur. Phys. J. D. 61 (2011) 241.6. K. Husimi, Proc. Phys. Math. Soc. Japan 22 (1940) 264.7. S.S. Mizrahi, Physica A 127 (1984) 241.8. S.S. Mizrahi, Physica A 135 (1986) 237.9. S.S. Mizrahi, Physica A 150 (1988) 541.10. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman,E.A. Cornell, Science 269 (1995) 198.11. K.B. Davis, M.O. Mewes, M.R. Andrews, N.J. van Druten,D.S. Durfee, D.M. Kurn, W. Ketterle, Phys. Rev. Lett. 75 (1995)3969.12. C.C. Bradley, C.A. Sackett, R.G. Hulet, Phys. Rev. Lett. 78(1997) 985.13. L. Mandel, Opt. Lett. 4 (1979) 205.14. E.C.G. Sudarshan, Phys. Rev. Lett. 10 (1963) 277.15. J. Bajer, A. Miranowicz, J. Opt. B 2 (2000) L10.16. C. Brif, Y. Ben-Aryeh, Quantum Op. 6 (1994) 391.17. M.O. Scully, M.S. Zubairy, Quantum optics (Cambridge Uni-versity Press, NY, 1997).18. H.J. Carmichael, Statistical methods in quantum optics, VolumeI (Springer, Berlin, 2010).Rev. Mex. Fis. E 60 (2014) 103–106

English

Different creation-destruction operators' ordering, quasi-probabilities, and Mandel parameter

EDUCATION Revista Mexicana de Fı́sica E60 (2014) 103–106 JULY–DECEMBER 2014Different creation-destruction operators’ ordering,quasi-probabilities, and Mandel parameterF. Penninia,b, A. PlastinoaaInstituto de F́ısica La Plata–CCT-CONICET, Universidad Nacional de La Plata,C.C. 727, 1900, La Plata, ArgentinabDepartamento de F́ısica, Universidad Cat́olica del Norte,Av. Angamos 0610, Antofagasta, Chile.Received 27 August 2014; accepted 27 October 2014In this work we provide an introductory discussion to quasi-probabilities in quantum optics and how to use them for evaluating the Mandelparameter.Keywords: Quantum distributions; phase space; Mandel parameter.En este trabajo proveemos una discusión introductoria sobre cuasi-probabilidades enóptica cúantica y las usamos para evaluar el parámetrode Mandel.Descriptores: Distribuciones cúanticas; espacio de fases; parámetro de Mandel.PACS: 03.67.-a; 05.30.-d; 42.50.-p; 42.50.Lc1. IntroductionThe focus of this article is the notion of quasi-probability.Why? because this is an important concept for quantum op-tics (among other fields). What is a quasi-probability dis-tribution (QPD)? It is a mathematical construction that, al-though resembling a probability distribution, does not exactlyfulfill the Kolmogorov’s axioms on which probability theoryis founded [1], more specifically, the third one may be vio-lated [1]. Quasi-probabilities exhibit general features of or-dinary probabilities. They yield expectation values with re-spect to the weights of the quasi-distribution. However, letus reiterate, they violate the third probability postulate [1], inthe sense that regions integrated under them do NOT rep-resent probabilities of mutually exclusive states. In somecases, quasi-probability distributions exhibit zones of nega-tive probability density. QPDs often arise in the phase spacerepresentation of quantum mechanics, customarily employedin quantum optics, time-frequency analysis, etc.It is well known that the dynamics of a quantum systemis determined by a master equation. One faces an equation ofmotion for the density operator̂ρ, expressed via a completeorthonormal basis. It is always possible to cast the densityoperator in a diagonal manner, provided that an overcompletebasis be used [2]. The most famous such basis is that of co-herent states|α〉 [3]. One writes [2]ρ̂ =∫d2απf(α, α∗) |α〉〈α|. (1)We see that a central role is assigned to the ordinary func-tion f , that becomes endowed with the features of a quasi-probability distribution. In particular, one hasd2α/π =dxdp/2π~. The system evolves as prescribed by the evolu-tion of the quasi-probability distribution function. Coherentstates, right eigenstates of the annihilation operatorˆ, serveas the overcomplete basis in such a build-up [2,3].In quantum optics f is called the functionP . One speaksof the P-representation. There exists two other importantrepresentations, known as theQ- andW - ones. There ex-ists a family of different representations, each connected toa different ordering of the underlying creation and destruc-tion operatorŝa and â†. Historically, the first of these isthe Wigner quasi-probability distributionW [4], related tosymmetric operator ordering. In quantum optics the particlenumber operator is naturally expressed in normal order and,in the pertinent scenario, the associated representation of thephase space distribution is the Glauber–SudarshanP o e [3].In addition to these two (W and P ), one often encountersother quasi-probability distributions emerging in alternativerepresentations of the phase space distribution [5]. A quitepopular representation is the HusimiQ one [6-9], employedwhen operators are in anti-normal order.In this paper we wish illustrate how these orderings aretackled so as to find theW , P , andQ representations for theimportant instance of the Harmonic Oscillator (HO) of angu-lar frequencyω in the canonical ensemble formulation. Insuch a scenario one deals with three functions associated tothe W, Q, and P representations that are simple Gaussians andthe treatment becomes entirely analytical, a very convenientdidactic feature. The HO is a really important system thatyields insights usually having a wide impact. Thus, the HOconstitutes much more than a mere simple example. Nowa-days, it is of particular interest for the dynamics of bosonic orfermionic atoms contained in magnetic traps [10-12] as wellas for any system that exhibits an equidistant level spacingin the vicinity of the ground state, like nuclei or Luttingerliquids. For the sake of concreteness we focus attention ona physical important quantity, a noise indicator, called theMandel parameter.So as to accomplish our didactic purposes, this commu-nication is organized as follows. In Sec. 2. we review some104 F. PENNINI AND A. PLASTINOnotions about the Mandel parameter. In Sec. 3. we dis-cuss the formulation of the Mandel parameter in terms oftheP−function for a thermal state. In Sec. 4. we reproducethe same ideas viaQ−function, and in Sec. 5. we recalculatethese results in terms of the Wigner function. Finally, someconclusions are drawn in Sec. 6.2. Mandel parameterSince the very beginnings of quantum mechanics there hasbeen interest in gauging how non-classical a quantum systemcan be. Several questions can be asked in this respect. Forinstance, how closely does the probability distribution for astate after a given series of measurements resemble that of aclassical system? Or, how much is a quantum state perturbedby a measurement on the system?A useful parameter to characterize non-classicality inquantum optics is the so-called Mandel parameter, introducedby Mandel in Ref. 13. It is defined asQM = (∆n̂)2〈n̂〉 − 1 ≡ F − 1, (2)and intimately linked to the normalized variance,which is also denominates the quantum Fano factorF=(∆n̂)2/〈n̂〉 [15] of the photon distribution, with(∆n̂)2 = 〈n̂2〉 − 〈n̂〉2, and the number operator̂n = â†â.One must take note that• ForF < 1 (QM ≤ 0), emitted light is referred to assub-Poissonian since it has photo-count noise smallerthan that of coherent (ideal laser) light with the sameintensity (F = 1; QM = 0), whereas• for F > 1, (QM > 0) the light is called super-Poissonian, exhibiting photo-count noise higher thanthe coherent-light noise.If for the photon number operator̂n the fluctuations in̂n dis-appear, the Mandel parameter becomesQM = −1 (F = 0).We pass now to our central issue, the evaluation of the Man-del parameter for our three quasi-probability instances,i. .,P , Q, andW .3. Mandel Parameter via theP−functionThe most general density operator is just a superpositionof projection operators, known as the Glauber-SudarshanP−representation [14]. One hasρ̂ =∫d2απP (α, α∗) |α〉〈α|, (3)where the functionP (α, α∗) plays the role of a probabilitydensity for the distribution of values ofα over the complexplane. A coherent state|α〉 is a specific kind of quantum state,the one that most resembles a classical state. It is applicableto the quantum harmonic oscillator, the electromagnetic field,etc., and describes a maximal kind of coherence and a clas-sical kind of behavior. The states|α〉 are normalized,i.e.,〈α|α〉 = 1, and they provide us with a resolution of the iden-tity operator∫d2απ|α〉〈α| = 1, (4)which is a completeness relation for the coherent states [3].The standard coherent states|α〉 for the harmonic oscillatorare eigenstates of the annihilation operatorâ, with complexeigenvaluesα, which satisfyâ|α〉 = α|α〉 [3]. Also, â, a†,andn̂ fulfills the commutation relations[â, â†] = 1, (5a)[n̂, â] = −â, (5b)[n̂, â†] = â†. (5c)As stated above,P (α) is a quasi-probability distributionfunction because it can have negative values and strong sin-gularities, especially when the density operator correspondsto a nonclassical state with sub-Poisson photon statistics(see Ref. 16 and references therein). When this functiontends to vary little over large ranges of the parameterα, thenonorthogonality of the coherent states will make little differ-ence, andP (α) can be interpreted as a probability distribu-tion [3]. The normalization property of the density operatorrequires thatP (α) obey the normalization condition [3]Tr ρ̂ =∫d2απP (α, α∗) = 1. (6)Accordingly, the expectation value of an observableÂ isgiven by [16]〈Â〉 = Tr(ρ̂Â) =∫d2απP (α, α∗) 〈α|Â|α〉. (7)In this context,the average particle-number–an importantquantity for our present considerations– acquires a simpleform that, according to Eq. (3), can be cast in the fashion [3]〈n̂〉 = Tr(ρ̂â†â) = 〈â†â〉P=∫d2απP (α, α∗) |α|2 = 〈|α|2〉P , (8)indicating that the average photon number is the meansquared absolute value of the amplitudeα. Note that〈 〉Pis the average with respect toP (α).In particular, for a thermal state for which the density op-erator is of the form prescribed by the canonical ensemble’srepresentation we haveρ̂ = (1− e−β~ω) e−β~ωâ†â (9)Rev. Mex. Fis.E 60 (2014) 103–106DIFFERENT CREATION-DESTRUCTION OPERATORS’ ORDERING, QUASI-PROBABILITIES, AND MANDEL PARAMETER 105and the correspondingP−function becomesP (α, α∗) =1〈n̂〉 exp(−|α|2〈n̂〉), (10)while the average particle-number is given by [3]〈n̂〉 = 〈|α|2〉P = e−β~ω1− e−β~ω . (11)Now, for all normal-ordered operator-averages we have〈â†râs〉P =∫d2απP (α, α∗)α∗rαs. (12)In particular, forr = s = 2, and taking into account thecommutation relations between̂a andâ†, we find〈â†2â2〉P = 〈(â†â)2〉P − 〈â†â〉P ≡ 〈n̂2〉 − 〈n̂〉, (13)where〈â†2â2〉P =∫d2απP (α, α∗) |α|4 = 〈|α|4〉P . (14)Considering Eqs. (13) and (14), we realize that one can write〈n̂2〉 = 〈|α|2〉P + 〈n̂〉. (15)In accordance with Eqs. (11) and (15), the Mandel pa-rameter now becomes, via theP−function,QM = 〈|α|4〉P − 〈|α|2〉2P〈|α|2〉P =e−β~ω1− e−β~ω , (16)a temperature dependent expression, where the statisticalaverages in phase space are computed utilizingP (α) as aweight function〈|α|s〉P = 2∞∫0dα |α|s+1 P (α, α∗), (17)with s = 2, 4.4. Mandel parameter via theQ−functionA closely related phase space distribution is obtained by tak-ing the diagonal matrix element of density operatorρ̂Q(α, α∗) = 〈α|ρ̂|α〉, (18)which it has all the properties of a classical probability dis-tribution [18]. For a thermal state, theQ−function is thegaussian quantity given byQ(α, α∗) =11 + 〈n̂〉 exp(− |α|21 + 〈n̂〉). (19)The Q−representation gives operator averages in antinor-mal order, so in this case the antinormal-ordered average be-comes [18]〈âsâ†r〉Q =∫d2απQ(α, α∗)α∗rαs, (20)where 〈 〉Q denotes the average with respect toQ(α, α∗).Takingr = s = 2 we have〈â2â†2〉Q =∫d2απQ(α, α∗) |α|4 = 〈|α|4〉Q. (21)Considering the identitŷa2â†2 = n̂2 + 3n̂ + 2 and Eq. (21),it is easy to show that〈n̂2〉 = 〈|α|4〉Q − 3〈n̂〉 − 2. (22)Also, for s = r = 1, Eq. (20) reduces to〈ââ†〉Q =∫d2απQ(α, α∗) |α|2 = 〈|α|2〉Q, (23)and we have〈n̂〉 = 〈|α|2〉Q − 1. (24)Finally, from Eqs. (22) and (24) the Mandel parameter interms of averages ofQ turns out to beQM =〈|α|4〉Q − 2〈|α|2〉Q − 〈|α|2〉2Q + 1〈|α|2〉Q − 1=e−β~ω1− e−β~ω , (25)and coincides with the pertinent expression obtained via theP -representation. Here, the means values are calculated us-ing〈|α|s〉Q = 2∞∫0dα |α|s+1 Q(α, α∗), (26)with s = 2, 4, with Q(α, α∗) given by (19).5. Mandel parameter via the Wigner functionThe Wigner function can be obtained from theP−functionfrom the relation [17]W (α, α∗) = 2∫d2zπP (z, z∗) exp(−2|α− z|2), (27)such that for a thermal state this function becomesW (α, α∗) =1〈n̂〉+ 1/2 exp(− |α|2〈n̂〉+ 1/2), (28)with 1/(〈n̂〉 + 1/2) = 2 tanh(β~ω/2). The symmetric or-dered operator used in this Wigner representation is, in thiscase,〈(â†râs)S〉W =∫d2απW (α, α∗) α∗rαs, (29)Rev. Mex. Fis.E 60 (2014) 103–106106 F. PENNINI AND A. PLASTINOwhere(â†râs)S denotes the symmetric operator product ofthe creation operators and annihilation ones [18], and〈 〉Windicates the average with respect toW (α, α∗). Thus, forr = s = 2 we get〈(â†2â2)S〉W =∫d2απW (α, α∗) |α|4=16(〈â†2â2〉+〈â†ââ†â〉+ 〈â†â2â†〉+ 〈ââ†2â〉+ 〈ââ†ââ†〉+ 〈â2â†2〉), (30)implying, after to use the commutation relations (5), that〈n̂2〉 = 〈|α|4〉W − 〈n̂〉 − 12 . (31)Also, for r = s = 1 we get,〈(â†â)S〉W =∫d2απW (α, α∗) |α|2=12(〈âa†〉+ 〈â†â〉), (32)yielding〈n̂〉 = 〈|α|2〉W − 12 . (33)From Eqs. (31) and (33) we obtain now the Mandel parame-ter in the fashionQM = 〈|α|4〉W − 〈|α|2〉2W − 〈|α|2〉W + 1/4〈|α|2〉W − 1/2=1− tanh(β~ω/2)2 tanh(β~ω/2)=e−β~ω1− e−β~ω , (34)where we have considered the mean values of|α|2 and|α|4according to〈|α|s〉W = 2∞∫0dα |α|s+1 W (α, α∗), (35)with s = 2, 4, beingW (α, α∗) a statistical wight functiongiven by (28).Interestingly enough, the Mandel parameter for a thermalstate is the same in the three representations we are discussingin this work.6. ConclusionsWe have shown into some detail how to proceed to evalu-ate the quasi-probability distributions corresponding, respec-tively, to the P-, Q- and Wigner representations in the case ofa thermal state for the harmonic oscillator. Using them wehave computed the noise-factor called the Mandel parameter.We have seen that it is the same quantity, independent on thenature of the quasi-probability distribution. It is hoped thatthese reflections may help students in their efforts to navigatequantum optics waters.AcknowledgementF. Pennini would like to thank partial financial support byFONDECYT, grant 1110827.This work was also partially supported by the projectPIP1177 of CONICET (Argentina), and the projectsFIS2008-00781/FIS (MICINN)-FEDER (EU) (Spain, EU).1. J. Von Plato,Grundbegriffe der Wahrscheinlichkeitsrechnungin Grattan-Guinness, I., (ed., Landmark Writings in WesternMathematics, Elsevier, Amsterdam, pp 960-69, 2005).2. E.C.G. Sudarshan,Phys. Rev. Lett.10 (1963) 277.3. R.J. Glauber,Phys. Rev.131(1963) 2766.4. E.P. Wigner,Phys. Rev.40 (1932) 749.5. F. Pennini, A. Plastino,Eur. Phys. J. D.61 (2011) 241.6. K. Husimi, Proc. Phys. Math. Soc. Japan22 (1940) 264.7. S.S. Mizrahi,Physica A127(1984) 241.8. S.S. Mizrahi,Physica A135(1986) 237.9. S.S. Mizrahi,Physica A150(1988) 541.10. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman,E.A. Cornell,Science269(1995) 198.11. K.B. Davis, M.O. Mewes, M.R. Andrews, N.J. van Druten,D.S. Durfee, D.M. Kurn, W. Ketterle,Phys. Rev. Lett.75(1995)3969.12. C.C. Bradley, C.A. Sackett, R.G. Hulet,Phys. Rev. Lett.78(1997) 985.13. L. Mandel,Opt. Lett.4 (1979) 205.14. E.C.G. Sudarshan,Phys. Rev. Lett.10 (1963) 277.15. J. Bajer, A. Miranowicz,J. Opt. B2 (2000) L10.16. C. Brif, Y. Ben-Aryeh,Quantum Op.6 (1994) 391.17. M.O. Scully, M.S. Zubairy,Quantum optics(Cambridge Uni-versity Press, NY, 1997).18. H.J. Carmichael,Statistical methods in quantum optics, VolumeI (Springer, Berlin, 2010).Rev. Mex. Fis.E 60 (2014) 103–106

Different creation-destruction operators ordering, quasi-probabilities, and Mandel parameter

CONICET Digital

EDUCATION Revista Mexicana de Fı´sica E 60 (2014) 103–106 JULY–DECEMBER 2014Different creation-destruction operators’ ordering,quasi-probabilities, and Mandel parameterF. Penninia,b, A. PlastinoaaInstituto de Fı´sica La Plata–CCT-CONICET, Universidad Nacional de La Plata,C.C. 727, 1900, La Plata, ArgentinabDepartamento de Fı´sica, Universidad Cato´lica del Norte,Av. Angamos 0610, Antofagasta, Chile.Received 27 August 2014; accepted 27 October 2014In this work we provide an introductory discussion to quasi-probabilities in quantum optics and how to use them for evaluating the Mandelparameter.Keywords: Quantum distributions; phase space; Mandel parameter.En este trabajo proveemos una discusio´n introductoria sobre cuasi-probabilidades en o´ptica cua´ntica y las usamos para evaluar el para´metrode Mandel.Descriptores: Distribuciones cua´nticas; espacio de fases; para´metro de Mandel.PACS: 03.67.-a; 05.30.-d; 42.50.-p; 42.50.Lc1. IntroductionThe focus of this article is the notion of quasi-probability.Why? because this is an important concept for quantum op-tics (among other fields). What is a quasi-probability dis-tribution (QPD)? It is a mathematical construction that, al-though resembling a probability distribution, does not exactlyfulfill the Kolmogorov’s axioms on which probability theoryis founded [1], more specifically, the third one may be vio-lated [1]. Quasi-probabilities exhibit general features of or-dinary probabilities. They yield expectation values with re-spect to the weights of the quasi-distribution. However, letus reiterate, they violate the third probability postulate [1], inthe sense that regions integrated under them do NOT rep-resent probabilities of mutually exclusive states. In somecases, quasi-probability distributions exhibit zones of nega-tive probability density. QPDs often arise in the phase spacerepresentation of quantum mechanics, customarily employedin quantum optics, time-frequency analysis, etc.It is well known that the dynamics of a quantum systemis determined by a master equation. One faces an equation ofmotion for the density operator ρˆ, expressed via a completeorthonormal basis. It is always possible to cast the densityoperator in a diagonal manner, provided that an overcompletebasis be used [2]. The most famous such basis is that of co-herent states |α〉 [3]. One writes [2]ρˆ =∫d2αpif(α, α∗) |α〉〈α|. (1)We see that a central role is assigned to the ordinary func-tion f , that becomes endowed with the features of a quasi-probability distribution. In particular, one has d2α/pi =dxdp/2pi~. The system evolves as prescribed by the evolu-tion of the quasi-probability distribution function. Coherentstates, right eigenstates of the annihilation operator aˆ, serveas the overcomplete basis in such a build-up [2, 3].In quantum optics f is called the function P . One speaksof the P-representation. There exists two other importantrepresentations, known as the Q- and W - ones. There ex-ists a family of different representations, each connected toa different ordering of the underlying creation and destruc-tion operators aˆ and aˆ†. Historically, the first of these isthe Wigner quasi-probability distribution W [4], related tosymmetric operator ordering. In quantum optics the particlenumber operator is naturally expressed in normal order and,in the pertinent scenario, the associated representation of thephase space distribution is the Glauber–Sudarshan P one [3].In addition to these two (W and P ), one often encountersother quasi-probability distributions emerging in alternativerepresentations of the phase space distribution [5]. A quitepopular representation is the Husimi Q one [6-9], employedwhen operators are in anti-normal order.In this paper we wish illustrate how these orderings aretackled so as to find the W , P , and Q representations for theimportant instance of the Harmonic Oscillator (HO) of angu-lar frequency ω in the canonical ensemble formulation. Insuch a scenario one deals with three functions associated tothe W, Q, and P representations that are simple Gaussians andthe treatment becomes entirely analytical, a very convenientdidactic feature. The HO is a really important system thatyields insights usually having a wide impact. Thus, the HOconstitutes much more than a mere simple example. Nowa-days, it is of particular interest for the dynamics of bosonic orfermionic atoms contained in magnetic traps [10-12] as wellas for any system that exhibits an equidistant level spacingin the vicinity of the ground state, like nuclei or Luttingerliquids. For the sake of concreteness we focus attention ona physical important quantity, a noise indicator, called theMandel parameter.So as to accomplish our didactic purposes, this commu-nication is organized as follows. In Sec. 2. we review some104 F. PENNINI AND A. PLASTINOnotions about the Mandel parameter. In Sec. 3. we dis-cuss the formulation of the Mandel parameter in terms ofthe P−function for a thermal state. In Sec. 4. we reproducethe same ideas via Q−function, and in Sec. 5. we recalculatethese results in terms of the Wigner function. Finally, someconclusions are drawn in Sec. 6.2. Mandel parameterSince the very beginnings of quantum mechanics there hasbeen interest in gauging how non-classical a quantum systemcan be. Several questions can be asked in this respect. Forinstance, how closely does the probability distribution for astate after a given series of measurements resemble that of aclassical system? Or, how much is a quantum state perturbedby a measurement on the system?A useful parameter to characterize non-classicality inquantum optics is the so-called Mandel parameter, introducedby Mandel in Ref. 13. It is defined asQM = (∆nˆ)2〈nˆ〉 − 1 ≡ F − 1, (2)and intimately linked to the normalized variance,which is also denominates the quantum Fano factorF=(∆nˆ)2/〈nˆ〉 [15] of the photon distribution, with(∆nˆ)2 = 〈nˆ2〉 − 〈nˆ〉2, and the number operator nˆ = aˆ†aˆ.One must take note that• For F < 1 (QM ≤ 0), emitted light is referred to assub-Poissonian since it has photo-count noise smallerthan that of coherent (ideal laser) light with the sameintensity (F = 1; QM = 0), whereas• for F > 1, (QM > 0) the light is called super-Poissonian, exhibiting photo-count noise higher thanthe coherent-light noise.If for the photon number operator nˆ the fluctuations in nˆ dis-appear, the Mandel parameter becomes QM = −1 (F = 0).We pass now to our central issue, the evaluation of the Man-del parameter for our three quasi-probability instances, i.e.,P , Q, and W .3. Mandel Parameter via the P−functionThe most general density operator is just a superpositionof projection operators, known as the Glauber-SudarshanP−representation [14]. One hasρˆ =∫d2αpiP (α, α∗) |α〉〈α|, (3)where the function P (α, α∗) plays the role of a probabilitydensity for the distribution of values of α over the complexplane. A coherent state |α〉 is a specific kind of quantum state,the one that most resembles a classical state. It is applicableto the quantum harmonic oscillator, the electromagnetic field,etc., and describes a maximal kind of coherence and a clas-sical kind of behavior. The states |α〉 are normalized, i.e.,〈α|α〉 = 1, and they provide us with a resolution of the iden-tity operator ∫d2αpi|α〉〈α| = 1, (4)which is a completeness relation for the coherent states [3].The standard coherent states |α〉 for the harmonic oscillatorare eigenstates of the annihilation operator aˆ, with complexeigenvalues α, which satisfy aˆ|α〉 = α|α〉 [3]. Also, aˆ, a†,and nˆ fulfills the commutation relations[aˆ, aˆ†] = 1, (5a)[nˆ, aˆ] = −aˆ, (5b)[nˆ, aˆ†] = aˆ†. (5c)As stated above, P (α) is a quasi-probability distributionfunction because it can have negative values and strong sin-gularities, especially when the density operator correspondsto a nonclassical state with sub-Poisson photon statistics(see Ref. 16 and references therein). When this functiontends to vary little over large ranges of the parameter α, thenonorthogonality of the coherent states will make little differ-ence, and P (α) can be interpreted as a probability distribu-tion [3]. The normalization property of the density operatorrequires that P (α) obey the normalization condition [3]Tr ρˆ =∫d2αpiP (α, α∗) = 1. (6)Accordingly, the expectation value of an observable Aˆ isgiven by [16]〈Aˆ〉 = Tr(ρˆAˆ) =∫d2αpiP (α, α∗) 〈α|Aˆ|α〉. (7)In this context, the average particle-number –an importantquantity for our present considerations– acquires a simpleform that, according to Eq. (3), can be cast in the fashion [3]〈nˆ〉 = Tr(ρˆaˆ†aˆ) = 〈aˆ†aˆ〉P=∫d2αpiP (α, α∗) |α|2 = 〈|α|2〉P , (8)indicating that the average photon number is the meansquared absolute value of the amplitude α. Note that 〈 〉Pis the average with respect to P (α).In particular, for a thermal state for which the density op-erator is of the form prescribed by the canonical ensemble’srepresentation we haveρˆ = (1− e−β~ω) e−β~ωaˆ†aˆ (9)Rev. Mex. Fis. E 60 (2014) 103–106DIFFERENT CREATION-DESTRUCTION OPERATORS’ ORDERING, QUASI-PROBABILITIES, AND MANDEL PARAMETER 105and the corresponding P−function becomesP (α, α∗) =1〈nˆ〉 exp(−|α|2〈nˆ〉), (10)while the average particle-number is given by [3]〈nˆ〉 = 〈|α|2〉P = e−β~ω1− e−β~ω . (11)Now, for all normal-ordered operator-averages we have〈aˆ†raˆs〉P =∫d2αpiP (α, α∗)α∗rαs. (12)In particular, for r = s = 2, and taking into account thecommutation relations between aˆ and aˆ†, we find〈aˆ†2aˆ2〉P = 〈(aˆ†aˆ)2〉P − 〈aˆ†aˆ〉P ≡ 〈nˆ2〉 − 〈nˆ〉, (13)where〈aˆ†2aˆ2〉P =∫d2αpiP (α, α∗) |α|4 = 〈|α|4〉P . (14)Considering Eqs. (13) and (14), we realize that one can write〈nˆ2〉 = 〈|α|2〉P + 〈nˆ〉. (15)In accordance with Eqs. (11) and (15), the Mandel pa-rameter now becomes, via the P−function,QM = 〈|α|4〉P − 〈|α|2〉2P〈|α|2〉P =e−β~ω1− e−β~ω , (16)a temperature dependent expression, where the statisticalaverages in phase space are computed utilizing P (α) as aweight function〈|α|s〉P = 2∞∫0dα |α|s+1 P (α, α∗), (17)with s = 2, 4.4. Mandel parameter via the Q−functionA closely related phase space distribution is obtained by tak-ing the diagonal matrix element of density operator ρˆQ(α, α∗) = 〈α|ρˆ|α〉, (18)which it has all the properties of a classical probability dis-tribution [18]. For a thermal state, the Q−function is thegaussian quantity given byQ(α, α∗) =11 + 〈nˆ〉 exp(− |α|21 + 〈nˆ〉). (19)The Q−representation gives operator averages in antinor-mal order, so in this case the antinormal-ordered average be-comes [18]〈aˆsaˆ†r〉Q =∫d2αpiQ(α, α∗)α∗rαs, (20)where 〈 〉Q denotes the average with respect to Q(α, α∗).Taking r = s = 2 we have〈aˆ2aˆ†2〉Q =∫d2αpiQ(α, α∗) |α|4 = 〈|α|4〉Q. (21)Considering the identity aˆ2aˆ†2 = nˆ2 + 3nˆ+ 2 and Eq. (21),it is easy to show that〈nˆ2〉 = 〈|α|4〉Q − 3〈nˆ〉 − 2. (22)Also, for s = r = 1, Eq. (20) reduces to〈aˆaˆ†〉Q =∫d2αpiQ(α, α∗) |α|2 = 〈|α|2〉Q, (23)and we have〈nˆ〉 = 〈|α|2〉Q − 1. (24)Finally, from Eqs. (22) and (24) the Mandel parameter interms of averages of Q turns out to beQM =〈|α|4〉Q − 2〈|α|2〉Q − 〈|α|2〉2Q + 1〈|α|2〉Q − 1=e−β~ω1− e−β~ω , (25)and coincides with the pertinent expression obtained via theP -representation. Here, the means values are calculated us-ing〈|α|s〉Q = 2∞∫0dα |α|s+1Q(α, α∗), (26)with s = 2, 4, with Q(α, α∗) given by (19).5. Mandel parameter via the Wigner functionThe Wigner function can be obtained from the P−functionfrom the relation [17]W (α, α∗) = 2∫d2zpiP (z, z∗) exp(−2|α− z|2), (27)such that for a thermal state this function becomesW (α, α∗) =1〈nˆ〉+ 1/2 exp(− |α|2〈nˆ〉+ 1/2), (28)with 1/(〈nˆ〉 + 1/2) = 2 tanh(β~ω/2). The symmetric or-dered operator used in this Wigner representation is, in thiscase,〈(aˆ†raˆs)S〉W =∫d2αpiW (α, α∗)α∗rαs, (29)Rev. Mex. Fis. E 60 (2014) 103–106106 F. PENNINI AND A. PLASTINOwhere (aˆ†raˆs)S denotes the symmetric operator product ofthe creation operators and annihilation ones [18], and 〈 〉Windicates the average with respect to W (α, α∗). Thus, forr = s = 2 we get〈(aˆ†2aˆ2)S〉W=∫d2αpiW (α, α∗) |α|4=16(〈aˆ†2aˆ2〉+〈aˆ†aˆaˆ†aˆ〉+ 〈aˆ†aˆ2aˆ†〉+ 〈aˆaˆ†2aˆ〉+ 〈aˆaˆ†aˆaˆ†〉+ 〈aˆ2aˆ†2〉), (30)implying, after to use the commutation relations (5), that〈nˆ2〉 = 〈|α|4〉W − 〈nˆ〉 − 12 . (31)Also, for r = s = 1 we get,〈(aˆ†aˆ)S〉W =∫d2αpiW (α, α∗) |α|2=12(〈aˆa†〉+ 〈aˆ†aˆ〉), (32)yielding〈nˆ〉 = 〈|α|2〉W − 12 . (33)From Eqs. (31) and (33) we obtain now the Mandel parame-ter in the fashionQM = 〈|α|4〉W − 〈|α|2〉2W − 〈|α|2〉W + 1/4〈|α|2〉W − 1/2=1− tanh(β~ω/2)2 tanh(β~ω/2)=e−β~ω1− e−β~ω , (34)where we have considered the mean values of |α|2 and |α|4according to〈|α|s〉W = 2∞∫0dα |α|s+1W (α, α∗), (35)with s = 2, 4, being W (α, α∗) a statistical wight functiongiven by (28).Interestingly enough, the Mandel parameter for a thermalstate is the same in the three representations we are discussingin this work.6. ConclusionsWe have shown into some detail how to proceed to evalu-ate the quasi-probability distributions corresponding, respec-tively, to the P-, Q- and Wigner representations in the case ofa thermal state for the harmonic oscillator. Using them wehave computed the noise-factor called the Mandel parameter.We have seen that it is the same quantity, independent on thenature of the quasi-probability distribution. It is hoped thatthese reflections may help students in their efforts to navigatequantum optics waters.AcknowledgementF. Pennini would like to thank partial financial support byFONDECYT, grant 1110827.This work was also partially supported by the projectPIP1177 of CONICET (Argentina), and the projectsFIS2008-00781/FIS (MICINN)-FEDER (EU) (Spain, EU).1. J. Von Plato, Grundbegriffe der Wahrscheinlichkeitsrechnungin Grattan-Guinness, I., (ed., Landmark Writings in WesternMathematics, Elsevier, Amsterdam, pp 960-69, 2005).2. E.C.G. Sudarshan, Phys. Rev. Lett. 10 (1963) 277.3. R.J. Glauber, Phys. Rev. 131 (1963) 2766.4. E.P. Wigner, Phys. Rev. 40 (1932) 749.5. F. Pennini, A. Plastino, Eur. Phys. J. D. 61 (2011) 241.6. K. Husimi, Proc. Phys. Math. Soc. Japan 22 (1940) 264.7. S.S. Mizrahi, Physica A 127 (1984) 241.8. S.S. Mizrahi, Physica A 135 (1986) 237.9. S.S. Mizrahi, Physica A 150 (1988) 541.10. M.H. Anderson, J.R. Ensher, M.R. Matthews, C.E. Wieman,E.A. Cornell, Science 269 (1995) 198.11. K.B. Davis, M.O. Mewes, M.R. Andrews, N.J. van Druten,D.S. Durfee, D.M. Kurn, W. Ketterle, Phys. Rev. Lett. 75 (1995)3969.12. C.C. Bradley, C.A. Sackett, R.G. Hulet, Phys. Rev. Lett. 78(1997) 985.13. L. Mandel, Opt. Lett. 4 (1979) 205.14. E.C.G. Sudarshan, Phys. Rev. Lett. 10 (1963) 277.15. J. Bajer, A. Miranowicz, J. Opt. B 2 (2000) L10.16. C. Brif, Y. Ben-Aryeh, Quantum Op. 6 (1994) 391.17. M.O. Scully, M.S. Zubairy, Quantum optics (Cambridge Uni-versity Press, NY, 1997).18. H.J. Carmichael, Statistical methods in quantum optics, VolumeI (Springer, Berlin, 2010).Rev. Mex. Fis. E 60 (2014) 103–106

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Different creation-destruction operators ordering, quasi-probabilities, and Mandel parameter

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