Newton-Leibniz integration for ket-bra operators in quantum mechanics (V)—Deriving normally ordered bivariate-normal-distribution form of density operators and developing their phase space formalism

We show that Newton-Leibniz integration over Dirac’s ket-bra projection operators with continuum variables, which can be performed by the technique of integration within ordered product (IWOP) of operators [Hong-yi Fan, Hai-liang Lu, Yue Fan, Ann. Phys. 321 (2006) 480], can directly recast density operators and generalized Wigner operators into normally ordered bivariate-normal-distribution form, which has resemblance in statistics. In this way the phase space formalism of quantum mechanics can be developed. The Husimi operator, entangled Husimi operator and entangled Wigner operator for entangled particles with different masses are naturally introduced by virtue of the IWOP technique, and their physical meanings are explained.     

Newton-Leibniz integration for ket-bra operators (II)—application in deriving density operator and generalized partition function formula

Via the route of applying Newton-Leibniz integration rule to Dirac’s symbolic operators, we show that the density operator e^−βH, where H is multi-mode quadratic interacting boson operators, is a mapping of symplectic transformation in the coherent state representation appearing in the form of non-symmetric ket-bra operator integration. By virtue of the technique of integration within an ordered product (IWOP) of operators, we deduce its normally ordered form which directly leads to the generalized partition function formula and the Wigner function. Some new representations, such as displacement-squeezing correlated squeezed coherent states, constructed by the IWOP technique, also bring convenience in deriving partition functions.     

ABCD rule for Gaussian beam propagation in the context of quantum optics derived by the IWOP technique

The development of technique of integration within an ordered product (IWOP) of operators extends the Newton-Leibniz integration rule, originally applying to permutable functions, to the non-commutative quantum mechanical operators composed of Dirac’s ket-bra, which enables us to obtain the images of directly mapping symplectic transformation in classical phase space parameterized by [A, B; C, D] into quantum mechanical operator through the coherent state representation, we call them the generalized Fresnel operators (GFO) since they correspond to Fresnel transforms in Fourier optics. Based on GFO we find the ABCD rule for Gaussian beam propagation in the context of quantum optics (both in one-mode and two-mode cases) whose classical correspondence is just the ABCD rule in matrix optics. The entangled state representation is used in discussing the two-mode case.     

Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations

Newton-Leibniz integration rule only applies to commuting functions of continuum variables, while operators made of Dirac’s symbols (ket versus bra, e.g., |q〉〈q| of continuous parameter q) in quantum mechanics are usually not commutative. Therefore, integrations over the operators of type |〉〈| cannot be directly performed by Newton-Leibniz rule. We invented an innovative technique of integration within an ordered product (IWOP) of operators that made the integration of non-commutative operators possible. The IWOP technique thus bridges this mathematical gap between classical mechanics and quantum mechanics, and further reveals the beauty and elegance of Dirac’s symbolic method and transformation theory. Various applications of the IWOP technique, including constructing the entangled state representations and their applications, are presented.     

On Fermionic Entangled State Representation and Fermionic Entangled Wigner Operator

By analogy with the bosonic bipartite entangled state we construct fermionic entangled state with the Grassmann numbers. The Wigner operator in the fermionic entangled state representation is introduced, whose marginal distributions are understood in an entangled way. The technique of integration within an ordered product （IWOP） of Fermi operators is used in our discussion.     

On Fermionic Entangled State Representation and Fermionic Entangled Wigner Operator

By analogy with the bosonic bipartite entangled state we construct fermionic entangled state with the Grassmann numbers. The Wigner operator in the fermionic entangled state representation is introduced, whose marginal distributions are understood in an entangled way. The technique of integration within an ordered product (IWOP) of Fermi operators is used in our discussion.     

Newton-Leibniz integration for ket-bra operators in quantum mechanics (IV)—Integrations within Weyl ordered product of operators and their applications

We show that the technique of integration within normal ordering of operators [Hong-yi Fan, Hai-liang Lu, Yue Fan, Ann. Phys. 321 (2006) 480-494] applied to tackling Newton-Leibniz integration over ket-bra projection operators, can be generalized to the technique of integration within Weyl ordered product (IWWOP) of operators. The Weyl ordering symbol is introduced to find the Wigner operator’s Weyl ordering form Δ(p,q) =  δ(p − P)δ(q − Q) , and to find operators’ Weyl ordered expansion formula. A remarkable property is that Weyl ordering of operators is covariant under similarity transformation, so it has many applications in quantum statistics and signal analysis. Thus the invention of the IWWOP technique promotes the progress of Dirac’s symbolic method.     

s-Parameterized phase space distribution for describing no counts registered on a photonic detector

By introducing the generalized Wigner operator for s-parameterized quasiprobability distribution and employing the technique of integration within ordered product (IWOP) of operators (normally ordered, Weyl ordered or antinormally ordered), we derive two new quantum-mechanical formulas for describing no counts registered on a photonic detector when a light field’s density operator ρ is known, one involves ρ’s s-parameterized distribution function, and the other involves ρ’s coherent state mean value, when these information is known then using the new formulas to calculate no-photocount would be convenient.     

Normally-Ordered Time Evolution Operator for Mass-Varying Harmonic Oscillator and Wigner Function of Squeezed Number State

For investigating dynamic evolution of a mass-varying harmonic oscillator we constitute a ket-bra integration operator in coherent state representation and then perform this integral by virtue of the technique ofintegration within an ordered product of operators. The normally orderedtime evolution operator is thus obtained. We then derive the Wigner functionof$u(t)| n>, where | n> is a Fock state, which exhibits a generalized squeezing, the squeezing effect is related to the varying mass with time.     

Optical wavelet-fractional squeezing combinatorial transform

By virtue of the method of integration within ordered product(IWOP)of operators we find the normally ordered form of the optical wavelet-fractional squeezing combinatorial transform(WFrST)operator.The way we successfully combine them to realize the integration transform kernel of WFr ST is making full use of the completeness relation of Dirac’s ket–bra representation.The WFr ST can play role in analyzing and recognizing quantum states,for instance,we apply this new transform to identify the vacuum state,the single-particle state,and their superposition state.     

Husimi Functions of Excited Squeezed Vacuum States

Based on the Husimi operator in pure state form introduced by Fan et al., which is a squeezed coherent state projector, and the technique ofintegration within an ordered product (IWOP) of operators, as well as theentangled state representations, we obtain the Husimi functions of theexcited squeezed vacuum states (ESVS) and two marginal distributions of theHusimi functions of the ESVS.     

Generalized two-mode thermal vacuum state for the Hamiltonian of a parametric amplifier derived by the IWOP method

In this paper, we employ the technique of integration within an ordered product of operators (IWOP) to derive a generalized two-mode thermal vacuum state (GTTVS) for the Hamiltonian of a parametric amplifier. Application of GTTVS in calculating the ensemble average is discussed. The result is in agreement with that obtained by using the generalized Hellmann-Feynmann theorem.     

量子光学态的Ridgelet变换

利用有序算符内积分技术推导出一个有用的双模算符正规乘积公式. 然后在量子力学框架下, 计算出相干态、特殊压缩相干态、中介纠缠态表象的Radon变换. 在此基础上, 通过选取“墨西哥帽”母小波函数, 分别分析了以上三种量子光学态的Ridgelet变换. 关键词： 有序算符内积分技术 Radon变换 Ridgelet变换     

Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac’s symbolic method

By virtue of the new technique of performing integration over Dirac’s ket–bra operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel–Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel operator (GFO) to correspond to the generalized Fresnel transform (GFT) in classical optics. We derive GFO’s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used. All these have confirmed Dirac’s assertion: “...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory”.     

Wigner Function and Phase Probability Distribution of q-Analogue of Squeezed One-Photon State

In this paper, in terms of the technique of integration within an ordered product （IWOP） of operators and the properties of the inverses of q-deformed annihilation and creation operators, normalizable q-analogue of the squeezed one-photon state, which is quite different from one introduced by Song and Fan [Int. 3. Theor. Phys. 41 （2002） 695], is constructed. Moreover, the Wigner function and phase probability distribution of q-analogue of the squeezed one-photon state are examined.     

Some Expressions for Legendre Polynomials and Jacobian Polynomials Gained by Virtue of the IWOP Technique

By virtue of the coherent state and the technique of integration within an ordered product (IWOP) of operators we derive some expressions, which are in differential forms, for Legendre polynomials and Jacobian polynomials.     

Normally Ordered Operator Realization of U(1/1) Supergroup and Super-Jaynes-Cummings Model

By making use of the technique ofintegration within an ordered product (IWOP) of operators and the bosonic and fermionic coherent states we derive the normally ordered operator realization of U(1/1) supergroup. Its application in calculating the superpartition function of the supersymmetric Jaynes-Cummings model is demonstrated. The antinormally ordered realization of U(1/1) is also obtained.     

Wigner function and tomogram of the Hermite polynomial state

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, this paper derives the Wigner function for the Hermite polynomial state (HPS). The tomogram of the HPS is calculated with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics.     

New Integration Transformation Connecting Coherent State and Entangled State

With the help of technique of integration within an ordered product (IWOP) of operators we find new integration transformation connecting the coherent state and the biparticle entangled state. We also point out that under this kind of integration transformation the direct product of two single-variable Hermite polynomials behaves quite different from the two-variable Hermite polynomials, in this way we show that the latter is intrinsic to the phase space of quantum entanglement. As a byproduct, some operator identities for theoretical quantum optics can also be neatly expressed in terms of the two-variable Hermite polynomials.