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Using the thermal-entangled state representations, we obtain the analytical evolution of quantum states simultaneously undergoing two kinds of quantum noises that can be represented as the infinitive Kraus operator-sum representation. We also prove the normalization and trace preservation of Kraus operators. Besides, we obtain the evolutions of Wigner distribution and photon number distribution of any quantum state when simultaneously passing through two kinds of quantum noises.

Based on the idea of transition from classical optics to quantum optics we deduce the natural expressions of optical beam splitter (BS) and 2-cascaded BS operators in coherent state representation and also obtain their normally ordered forms. Moreover, we theoretically prepare some desired quantum entangled states and investigate several carried experiments. Finally, our work shall be focused on the following two aspects: the quantum-optical catalysis of the BS and quantum state truncation of the 2-cascaded BS.

Operators’ ordering is often employed in almost all calculations of quantum optics. In this paper by virtue of differentiation within ordered product of operators we put forward general differential formalism of normally and anti-normally ordering Bose operators, which seems neat and concise

Phase space analysis of quantum states is a newly developed topic in quantum optics. In this work we present Wigner phase space distributions for the two-mode binomial state produced by quantum entanglement between a vacuum state and a number state in a beamsplitter. By using two new binomial formulas involving two-variable Hermite polynomials and the so-called entangled Wigner operator, we find that the analytical Wigner function for the binomial state |ξ〉q ≡ D(ξ) |q, 0〉 is related to a Laguerre polynomial, i.e., $ W\left (\sigma _{,}\gamma \right ) =\frac {(-1)^{q}e^{-\left \vert \gamma \right \vert ^{2}-\left \vert \sigma \right \vert ^{2}}}{\pi ^{2}}L_{q}\left (\left \vert \frac {-\varsigma (\sigma -\gamma )+\sigma ^{\ast }+\gamma ^{\ast }} {\sqrt {1+|\varsigma |^{2}}}\right \vert ^{2}\right ) $ and its marginal distributions are proportional to the module-square of a single-variable Hermite polynomial. Also, the numerical results show that the larger number sum q of two modes lead to the stronger interference effect and the nonclassicality of the states |ξ〉q is stronger for odd q than for even q.

Exploiting the thermo entangled state approach, we successfully solve the master equation for describing the single-mode cavity driven by an oscillating external field in the heat reservoir and then get the analytical time-evolution rule for the density operator in the infinitive Kraus operator-sum representation. It is worth noting that the Kraus operator Ml, m is proved to be a trace-preserving quantum operation. As an application, the time-evolution for an initial coherent state ρ|β〉 = |β〉〈β| in such an environment is investigated, which shows that the initial coherent state decays to a new mixed state as a result of thermal noise, however the coherence can still be reserved for amplitude damping.

Using the technique of integration within an ordered product (IWOP) of operators we derive a new kind of bipartite entangled state |β 1,β 2〉, which is perfectly different from the preceding entangled states in construction. We also present main properties, generation scheme of |β 1,β 2〉, superposition of |β 1,β 2〉, Wigner function and some applications of |β 1,β 2〉 in quantum optics.

The new intermediate entangled state |η;θ〉 is proposed by virtue of IWOP technique, which is the common eigenvector of $[(\hat{x}_{1} - \hat{x}_{2})\cos\theta -(\hat{p}_{1} - \hat{p}_{2})\sin\theta ]$ and $[(\hat{x}_{1} +\hat{x}_{2})\sin\theta + (\hat{p}_{1} + \hat{p}_{2})\cos\theta ]$ . The squeezing transformation operator, Hadamard transformation operator, Fresnel transformation operator and Radon transform operator are constructed by |η;θ〉.

We first deduce the s-ordered expansion of the Wigner operator. Since Radon transformation of Wigner operator is just the intermediate representation |x〉 λ,ν projector, we naturally obtain the s-ordered product of |x〉 λ,νλ,ν〈x|. Accordingly, the completeness relation is still preserved under the s-ordering. Finally, based on it, we obtain the s-ordered expansion of some useful operator in quantum optics, and some new operator identities are revealed accordingly.

A new kind of bipartite coherent-entangled state (CES) is introduced in the two-mode Fock space which exhibits the properties of both coherent state and entangled state. The set of CESs make up a complete and partly nonorthogonal representation. A simple experimental scheme to produce the CES is proposed by using a beam splitter. Some applications of the CES are also presented in quantum optics.

The new coherent-entangled state |z,x;θ〉 is proposed in the two-mode Fock space, which exhibits both the properties of coherent and entangled states. The completeness relation of |z,x;θ〉 is proved by virtue of the technique of integral within an ordered product of operators. A generalized Hadamard-Fresnel complementary transformation derived by virtue of the coherent-entangled state |z,x;θ〉, which is unitary. The new unitary operator plays the role of both Hadamard transformation for \((\hat{a}_{1}\sin\theta -\hat{a}_{2}\cos\theta)\) and Fresnel transformation for \((\hat{a}_{1}\cos\theta +\hat{a}_{2}\sin\theta)\), respectively.

This paper examines the quantization for the capacitance coupling inductance-capacitance (LC) mesoscopic circuit. By virtue of the “invariant eigen-operator” (IEO) method, it deduces the selection rules of energy-level transition when the system is disturbed by the external electromagnetic field.

A new method of quantizing the mesoscopic RLC circuit is proposed, i.e., such a circuit can be equivalent to a changing mass simple harmonic oscillator. By virtue of the thermal entangled state representation and the generalized Hellmann-Feynman Theorem, the thermal effect for the system is investigated.

In this paper we obtain the Wigner functions of two-variable Hermite polynomial states (THPS) and their marginal distribution using the entangled state |ξ〉 representation. Also we obtain tomogram of THPS by virtue of the Radon transformation between the Wigner operator and the projection operator of another entangled state |η,τ1,τ2〉.

Using the entangled state representation of Wigner operator and some formulae related to the two-variable Hermite polynomials, the Wigner function of the squeezed pair coherent state (SPCS) and its two marginal distributions are derived. Based on the entangled Husimi operator introduced by Fan et al. (Phys. Lett. A 358:203, 2006) and the Weyl ordering invariance under similar transformations, we also obtain the Husimi function of the SPCS and its marginal distribution functions. The comparison between the two quasibability functions shows that, for the same amount of information included in two functions, the solving process of the Husimi function is simpler than that of the Wigner function.

The Hamilton operator for the mesoscopic LC circuits including complicated coupling is given. Because the practical circuits are inevitably influenced by the external environment, the effect of temperature on the quantum effects for the mesoscopic circuit must be taken into account. Then we discuss the thermal effect of the system by simulating it with the biphotonic process and the contribution of each process to the energy ensemble average by virtue of the generalized Hellmann-Feynman Theorem (GHFT).

By means of the Weyl correspondence and Wigner theorem the marginal distribution of Wigner function in mesoscopic RLC circuit at finite temperature was discussed. Here we endow the Wigner function with a new physical meaning, i.e., its marginal distributions’ statistical average for q2/(2C) and p2/(2L) are the temperature-related energy stored in capacity and in inductance of the mesoscopic RLC circuit, respectively.

Via the Hamilton dynamical approach we have constructed Hamiltonian for the mutual inductance coupling magnetic flux qubits. The entangled state representation is used to propose Cooper-pair number-phase quantization and the Hamiltonian operator for the whole system. The dynamical evolution of the phase difference operator and the Cooper-pairs number operator is investigated by virtue of Heisenberg equations.

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, the Wigner functions of the even and odd negative binomial states (NBSs) are obtained. We concentrate our analysis on the fact that the even and odd NBSs interpolate between the even and odd coherent states and the even and odd quasi-thermal states depending on the values of the parameters involved. Moreover, the tomograms of the even and odd NBSs are calculated by virtue of intermediate coordinate-momentum representation in quantum optics.

Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, the Wigner functions of the Klauder-Perelomov coherent states (KP-CSs) for the pseudoharmonic oscillator (PHO) are obtained and the variations of the Wigner functions with the parameters k and z are discussed. Moreover, the tomograms of the KP-CSs for the PHO are calculated by virtue of intermediate coordinate-momentum representation in quantum optics.

The time evolvement of quantum state has been researched by the schemes of Louisell’s common canonical quantization and Gauss Propagator for mesoscopic coupled double resonance circuit with electrical sources. Moreover, the formulae of time evolvement wave function and transition probability of quantum state of this circuit are given. It is more significant for the quantitative analysis quantum character of mesoscopic circuit.

The number-phase quantization scheme of the capacitance coupling double Josephson junction mesoscopic circuit is given. The eigenvalues and the eigenstates of the system are investigated. It is found that using this system the entangled states can be prepared.

We have studied the tunneling and fluctuations of a dipolar Bose–Einstein condensate in an optical lattice, it is found that there exist the tunneling and fluctuations between lattices l and l+1, l and l−1, respectively. In particular, when the optical lattice is infinitely long and the spin excitations are in the long-wavelength limit, tunneling effects disappear between lattices l and l+1, and that l and l−1, in this case the fluctuations are a constant, and the magnetic soliton appears.

In this paper we shall introduce two q-analogues of the squeezed states in terms of the technique of integration within an ordered product of operators and the properties of the inverses of q-deformed annihilation and creation operators, and some nonclassical properties of the states are examined. Furthermore, we obtain some new completeness relations composed of the bra and ket which are not mutually Hermitian conjugate.

The quantization scheme of a single-qubit structure with Superconducting Quantum Interference Device (SQUID) is given. By introducing a unitary matrix and by means of spectral decomposition, the Hamiltonian operator of the system is exactly formulated in compact forms in spin-1/2 notation. The eigenvalues and eigenstates of the system are discussed.

Wigner function in phase space has its physical meaning as marginal probability distribution in coordinate space and momentum space respectively, here we endow the Wigner function with a new physical meaning, i.e., its marginal distributions’ statistical average for q2/(2C) and p2/(2L) are the energy stored in capacity and in inductance of a mesoscopic L-C circuit at finite temperature, respectively.

By using of the invariant theory, we have studied phase of Bose-Einstein condensate in a double-well potential interacting with a time-dependent single-mode traveling-wave laser field, the dynamical and geometric phases are presented respectively. The Aharonov-Anandan phase is also obtained under the cyclical evolution.

Mesoscopic damped double resonance mutual capacitance coupled RLC circuit is quantized by the method of damped harmonic oscillator quantization. The Hamiltonian is diagonalized by unitary transformation. The eigenenergy spectra of this circuit are given. The quantum fluctuations of the charges and current of each loop are researched in excitation state of the squeezed vacuum state, the squeezed vacuum state and in vacuum state. It is show that, the quantum fluctuations of the charges and current are related to not only circuit inherent parameter and coupled magnitude, but also quantum number of excitation, squeezed coefficients, squeezed angle and damped resistance. And, because of damped resistance, the quantum fluctuation decay along with time.

The mesoscopic quartz piezoelectric crystal equivalent circuit is quantized by the method of damped harmonic oscillator quantization. It is shown that, when each branch is in the thermal vacuum states, the quantum fluctuations of the voltage, and current of each loop relate with not only the equivalent circuit inherent parameter, but also the temperature and decay according to exponent along with time.

A class of even and odd nonlinear coherent states are introduced. The properties of some related states, including quadrature squeezing, antibunching effect and phase probability distribution, are studied.

A class of k-quantum nonlinear coherent states, i.e., the k eigenstates of the powers B-k ( k greater than or equal to 3) of the annihilation operator B = a 1/f(N) of f-oscillators, is obtained and its completeness is investigated. An alternative method to construct them is proposed. We introduce a new kind of higher-order squeezing and sub-Poissonian distribution. The quantum statistical properties of the k states are studied. The result shows that all of the eigenstates can be generated by a linear superposition of k Roy-type nonlinear coherent states. These states may form a complete Hilbert space, and the M-th order [M = (n + 1/2) k; n = 0, 1,...] squeezing effects exist in all of the k states when k is even. There is the sub-Poissonian distribution in all of the states.

Using the Pegg–Barnett formalism of phase operator, we obtain phase probability distributions of new even and odd nonlinear coherent states. It is shown that the distributions for the states are rather different, and unlike the case of ordinary even and odd coherent states the Pegg–Barnett distribution clearly reflects the different character of quantum interference in the case of the new even and odd coherent states.

In this paper, the completeness of the k orthonormalized eigenstates of the operator (a(q) f(N-q))(k)(k greater than or equal to 3) is proved. We introduce a new kind of higher order squeezing and an antibunching. The properties of the Mth-order squeezing and the antibunching effect of the k states are investigated. The result shows that these states may form a complete Hilbert space, and the Mth order [M = (m + 1/2)k; m = 0, 1, 2....] squeezing effects exist in all of the k states when k is even. There is the antibunching effect in all of the states.

Starting from the classical equation of motion for a mesoscopic capacitance coupling circuit, we study the quantum fluctuations of charge and current of the mesoscopic capacitance coupling circuit in a displaced squeezed Fock state. It is found that the quantum fluctuations of charge and current in each component circuit depend on the devices of two circuits and the squeezing parameters, while the fluctuations do not depend on displacement parameters.

Starting from the equation of motion of a mesoscopic inductance coupling circuit, the quantum fluctuations of charge and current in the circuit are investigated in both the eigenstates of the system and the squeezed vacuum state. The results show that there exist quantum fluctuations of the charge and current in both cases, and the fluctuations in each component circuit are connected.

In this paper the N eigenstates of the Nth powers (N greater than or equal to 3) of the annihilation operator of an anharmonic oscillator are constructed and their completeness is investigated. We introduce a new kind of higher order squeezing, Mth-order SU(I, 1) squeezing. The properties of Mth-order SU(I, I) squeezing and anticorrelation of the N states are studied. The result show that these states may form a complete Hilbert space; the Mth-order [M = (n + 1/2)N; n = 0,1,...] SU(1, 1) squeezing effects exist in all of the N states when N is even. There is anticorrelation in all of them.

Starting from the equation of motion of anactive RLC circuit, the quantum effects of charge andcurrent in the mesoscopic circuit (RLC circuit) in thesqueezed vacuum state are investigated.