Time evolution of angular momentum coherent state derived by virtue of entangled state representation and a new binomial theorem

We study how can an angular momentum coherent state |τ> keeps its form-invariant during time evolution governed by the Hamiltonian H = f(t)J++ f^*(t)J-+ g(t)Jz. We discuss this topic in the context of boson realization of |τ>. By employing the entangled state representation |ζ> and deriving a new binomial theorem involving two-subscript Hermite polynomials, we derive the wave function <ζ|τ>, which turns out to be a single-subscript Hermite polynomial. Based on this result the maintenance of angular momentum coherent state during time evolution is examined, and the value of τ(t) is totally determined by the parameters involved in the Hamiltonian.

Radiating frequency of three-loop mesoscopic LC circuit with mutual inductance obtained by IEO method

Instead of normally tackling electric circuits by virtue oI the Klrctllaott＇s theorem wnose aim is to uerlvc voxt^gc, electric current, and electric impedence, our aim in this paper is to derive the characteristic frequency of a three-loop mesoscopic LC circuit with three mutual inductances, e.g., for the radiating frequency of the three-loop LC oscillator, we adopt the invariant eigen-operator （lEO） method to realize our aim.

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 ex- plore 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, de- riving 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 opera- tor （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 op- tics 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＂.

Evolution of the single-mode squeezed vacuum state in amplitude dissipative channel

用作为主人方程的一个答案导出密度操作员的不定式和表示的方法描述振幅由纠缠的州的表示的优点的消散的隧道，我们明白地显示出一个单个模式的挤压的真空状态的起始的密度操作符怎么演变为结果是一个挤压的混乱状态与的一个明确的混合状态压榨减少并且 decoherence 。我们调查平均光子数字，为这的光子统计分布混合了状态。

On the core of the fractional Fourier transform and its role in composing complex fractional Fourier transformations and Fresnel transformations

By a quantum mechanical analysis of the additive rule Fa [Fβ[f]] = Fα＋β[f], which the fractional Fourier transformation （FrFT） Fα [f] should satisfy, we reveal that the position-momentum mutual- transformation operator is the core element for constructing the integration kernel of FrFT. Based on this observation and the two mutually conjugate entangled-state representations, we then derive a core operator for enabling a complex fractional Fourier transformation （CFrFT）, which also obeys the additive rule. In a similar manner, we also reveal the fractional transformation property for a type of Fresnel operator.

On the entangled fractional squeezing transformation

We propose an entangled fractional squeezing transformation （EFrST） generated by using two mu- tually conjugate entangled state representations with the following operator： e-iα（a1a2＋a1a2）eiπa2a2; this transformation sharply contrasts the complex fractional Fourier transformation produced by e-ia（a1a1＋a2a2）eiπa2a2 （see Front. Phys. DOI 10.1007/s11467-014-0445-x）. The EFrST is obtained by converting the triangular functions in the integration kernel of the usual fractional Fourier transformation into hyperbolic functions, i.e., tan α→ tanh α and sin α→ sinh α. The fractional property of the EFrST can be well described by virtue of the properties of the entangled state representations.

Evolution law of Wigner function in laser process

Based on the density operator＇s operator-sum representation recently obtained by Fan and Hu for a laser process （Opt. Commun., 2008, 281： 5571; Opt. Commun., 2009, 282： 932; Phys. Lett. B, 2008, 22： 2435）, we derive the evolution law of Wigner operator, the law is concisely expressed in T exp[-2T（ate-（k-g）t - a.）（ae-（k-g）t a）]; where g the normally ordered form A（a,a＊,t） = -T/π., and a are the cavity gain and the loss, respectively, and T - （a - g）（g-t-g - 2ge-2（k-g）t）-1. When 1 exp[-2（at a＊）（a a）] which is the initial Wigner operator. Using this t = O, A（a,a＊,t） →1/π , formalism the evolution law of Wigner functions in laser process can be directly obtained.