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The development of quantum optics theory based on the method of integration within an ordered product of operators (IWOP) has greatly stimulated the study of quantum states in the light field, especially non-Gaussian states with various non-classical properties. In this paper, the two-mode squeezing operator is derived with integral theory within the Weyl ordering product of operators using a combinatorial field in which one mode is a chaotic field and the other mode is a vacuum field. The density operator of the new light field, its entanglement property and photon number distribution are analyzed. We also note that tracing a three-mode pure state can yield this new light field. These methods represent a theoretical approach to investigating new density operators of light fields.

In our previous papers, the classical fractional Fourier transform theory was incorporated into the quantum theoretical system using the theoretical method of quantum optics, and the calculation produced quantum mechanical operators corresponding to the generation of fractional Fourier transform. The core function of the coordinate–momentum exchange operators in the addition law of fractional Fourier transform was analyzed too. In this paper, the bivariate operator Hermite polynomial theory and the technique of integration within an ordered product of operators (IWOP) are used to establish the entanglement fractional Fourier transform theory to the extent of quantum. A new function generating formula and an operator for generating quantum entangled fractional Fourier transform are obtained using the fractional Fourier transform relationship in a pair of conjugated entangled state representations.

It is an important topic of the coupling between quantum and classical light fields. In this paper, a classical Gaussian light field $${e}^{-\frac{{q}^{2}}{2{\tau }_{1}^{2}}-\frac{{p}^{2}}{2{\tau }_{2}^{2}}}$$ is used to modulate the translational squeezed quantum light field. By using the integral method in the ordered operator, we find that the translational squeezed quantum light field modulated by Gaussian light is still a translational squeezed light, and its quantum properties remain unchanged, but chaotic parameters and compression parameters are changed. We also derive the formula of the Wigner function depending on the modulation parameters. It can be concluded that the integral method in the ordered operator is an effective way to deal with the mixing of classical and quantum light fields.

We find that the quadrature measurement theory through symmetric beamsplitter can be elaborated more precisely and more directly by virtue of the IWOP technique. Both the normally ordered and anti-normally ordered generating photocurrent operator for mements of photocurrent probability distribution are derived, and their applications are respectively presented.

In this paper we discuss the temperature variation of a new chaotic optical field after it passes through a single-mode quantum dissipation and diffusion channel respectively. Using the entangled state representation and the technique of integration within ordered product (IWOP) of operators, we solve the master equations and obtain the new output fields. We show the final explicit forms of the new output fields by detailed derivation. By virtue of thermo dynamics theory at finite temperature, we finally conclude the results of temperature variation of chaotic optical field both in quantum dissipation channel and diffusion channel.

In this paper we present a deduction of the Hellmann-Feynman (HF) theorem for the lowest eigenenergy $E_{0}\left (\lambda \right ) $ of a Hamiltonian $ H\left (\lambda \right ) $, that is : its second-order derivative with respect to he parameter $\lambda ,\frac {\partial ^{2}E_{0}}{\partial \lambda ^{2}},$ is always less than the expectation value of $\frac {\partial ^{2}H\left (\lambda \right ) }{\partial \lambda ^{2}}$ in the ground state. We also point out that the above deduction does not hold for the FH theorem in ensemble average. The electric polarizability of molecules is studied by the deduction of the HF theorem

For developing the phase space quantum mechanics we explore how to construct different masses’ entangled state representation with the use of tthe Weyl-Wigner correspondence. We shall adopt a new approach, which uses the Radon transform of two-mode Wigner operator in normally ordering form to realize our goal. As an application of the entangled state representation the new normally ordered generalized squeezing operator is derived. The technique of integration within ordered product of operators is fully used.

There have been numerous researches on spin chain while most are gathered in the thermal entanglement or the intrinsic decoherence of spin chain, and only few papers investigate the spin chain under dephasing channel. Thus the dynamics of open spin chain in thermal noises is desiderated to research. In this paper, the entanglement dynamics induced by 2-qubit anisotropy Heisenberg model that each qubit coupled with reservoir is investigated. The corresponding master equation is analytically solved by the new Ket-Bra Entangled State (KBES) method with computer; The evolution of entanglement is also analyzed in detail when initial state is described by Extended Wenger-Like(EWL) state. It’s found that the spin coupling can lead to the revival of entanglement; Besides the fluctuation frequency show a positive correlation with the coupling coefficient. The most distinguished discovery is that the system still keep part entangled even for time tend to infinity, which imply that people can inhibit the disentanglement process by internal interaction of system.

We construct a new state via thermo excitation on the thermo vacuum state. It is interestingly found that partial tracing over thermo-freedom will lead to a negative-binomial state characteristic of l in the a-mode space and annihilating one a-mode photon results in increasing one $\tilde {a}$-mode photon, which exhibits quantum entanglement between system and environment. We employ the summation method within ordered product of operators to proceed our discussion.

For a laser channel governed by the master equation of the density operator which involves both photon gain and loss parameters, g and κ, we derive a new formula for the photocount distribution at time t for an initial density operator ρ0. The formula brings convenience, since we need not derive the density operator ρ t at time t. Moreover, this formula is applied to the case of a damping channel and a diffusion channel, respectively.

We explore time evolution of squeezed chaotic state (SCS) in amplitude dissipation channel with damping coefficient κ. By using the technique of integration within ordered product (IWOP) of operators we recast the initial density operator ρ 0 of SCS into the form of normally ordered Gaussian operator with two definite variances σ i ( i = 1, 2), which is related to the average photon number of the chaotic field and the squeezing parameter. We find that ρ 0 evolves to ρ t keeping the Gaussian form invariant except for σ i 2 → σ ′ i 2 = 1 − ( 1 − σ i 2 ) e − 2 κ t , which tells that the variance can be characteristic of the damping of SCS. Based upon the form of σ ′ i 2 we further describe the damping of SCS in terms of the damping rule for squeezing parameter and the mean photon number of chaotic field.

In order to entangle the functions to be transformed, we proposed the entangled. Fourier integration transformation (EFIT) which has the property of keeping modulus-invariant for its inverse transformation. Then we then studied Wigner operator’s EFIT and found that a function’s EFIT is just related to its Weyl-corresponding operator’s matrix element, in so doing we also derived new operator re-ordering formulas $$ \delta \left(x-P\right)\left(y-Q\right)=\frac{1}{\pi }{\displaystyle \begin{array}{c}:\\ {}:\end{array}}{e}^{-2i\left(P-x\right)\left(Q-y\right)}\ {\displaystyle \begin{array}{c}:\\ {}:\end{array}} $$ ; $$ \delta \left(y-Q\right)\left(x-P\right)=\frac{1}{\pi }{\displaystyle \begin{array}{c}:\\ {}:\end{array}}{e}^{2i\left(P-x\right)\left(Q-y\right)}\ {\displaystyle \begin{array}{c}:\\ {}:\end{array}} $$ , where P, Q are momentum and coordinate operator respectively, the symbol $$ {\displaystyle \begin{array}{c}:\\ {}:\end{array}}\ {\displaystyle \begin{array}{c}:\\ {}:\end{array}} $$ denotes Weyl ordering. By virtue of EFIT we also found the operator which can generate fractional squeezing transformation.

Based on the infinite sum solution of the master equation corresponding to the diffusion channel we derive the law of time evolution of a two-mode squeezed vacuum state in a single-mode (say a -mode) diffusion channel. We find that the squeezing effect decreases, while two different chaotic field modes emerge. After t time period of the diffusion, the a -mode photon number is larger than b -mode photon number by t κ , where κ is the diffusion coefficient, thus by adjusting κ , photon number can controlled. Therefore, the diffusion process can be considered a quantum controlling scheme through photon addition.

We analyse the mesoscopic two-loop LC circuit in which a capacitance C is sharing and mutual inductance M exists between two inductances L1, L2. Instead of using the usual Kirhoff's voltage law to examine the equivalent inductance, we solve the Schrodinger equation for this quantized two-loop LC circuit to find the wave function in entangled state representation (ESR). In so doing, we derive the equivalent inductance L 1 L 2 − M 2 L 1 + 2 M + L 2 , and the quantized energy level formula E n = L 1 I 1 + M I 1 + I 2 + L 2 I 2 2 2 L 1 + 2 M + L 2 + n + 1 2 L 1 + 2 M + L 2 L 1 L 2 − M 2 C , (n = 0, 1, …) where the term L 1 I 1 + M I 1 + I 2 + L 2 I 2 2 2 L 1 + 2 M + L 2 , as comparing the common energy stored in an inductance LI 2 2 , is a newly found magnetic energy stored in the two-loop circuit while quantum radiation with the characteristic frequency L 1 + 2 M + L 2 L 1 L 2 − M 2 C taking place. We do this by virtue of the ESR method, since quantum entanglement arises from both the mutual inductance coupling and the capacitance in sharing, and using ESR is convenient.

We study the one-dimensional Schrödinger equation for the Hamiltonian involving the bipartite hard-core potential, [Formula: see text]. We select the suitable unitary transformation and employ the entangled state representation |η⟩, η = η1 + iη2, to find the quantization condition for energy E, [Formula: see text] which seems new.

We investigate how an optical Gaussian-enhenced chaotic state (GECS) evolves in an amplitude dissipation channel. We have used the integration within ordered product of operators technique to derive its evolution law and show that the dacay of photon number. Also both the normally ordered and anti-normally ordered of the evolution law are given.

Partial transpose is an important operation for quantifying the entanglement, here we study the (partial) transpose of any single (two-mode) operators. Using the Fock-basis expansion, it is found that the transposed operator of an arbitrary operator can be obtained by replacement of a^{{\dag}}(a) by a(a^{{\dag}}) instead of c-number within normal ordering form. The transpose of displacement operator and Wigner operator are studied, from which the relation of Wigner function, characteristics function and average values such as covariance matrix are constructed between density operator and transposed density operator. These observations can be further extended to multi-mode cases. As applications, the partial transpose of two-mode squeezed operator and the entanglement of two-mode squeezed vacuum through a laser channel are considered., Comment: 10 pages, no figure

We study quantum-mechanical diffusion master equation by comparison with the classical diffusion equation, and deduce how a pure coherent state evolves into a mixed state by diffusion both in classical way and in quantum mechanical way.

We examine the quantum effect of cooling down the environment temperature of mesoscopic LC circuit, and find that the ground state of the circuit is no longer in the thermo vacuum state, but in a negative binomial state. We calculate energy of the circuit in this new state, which increases with the cooling of the environment.

In this paper, we investigate how a m -photon enhanced chaotic field (PACF), denoted by 1 − e − λ m + 1 a † m e − λ a † a a m / m !, evolves in an amplitude damping channel with a damping constant κ . By using the technique of integration within ordered product of operators (IWOP), we find that it evolves into a Laguerre function-weighted chaotic field which is described by ⋮ e 1 − e λ e 2 κ t aa † L m e λ + 2 κ t aa † ⋮ , which is in antinormal ordering. Time evolution of photon number average and fluctuation in this process are also derived. The corresponding evolution law of the Wigner function is also deduced analytically.

For developing quantum mechanics theory in phase space, we expolre how does an initial density operator ρ 0’s classical Weyl correspondence function h 0 evolve into h t in a damping channel. We derive the following integration transformation that relating them $h_{t}\left (\alpha ,\alpha ^{\ast } \right ) =\frac {2}{T}\int \frac {d^{2}\beta } {\pi } h_{0}\left (\beta ,\beta ^{\ast } \right ) \exp \left [ -\frac {2}{T} \left (\alpha ^{\ast } -\beta ^{\ast } e^{-\kappa t}\right ) \left (\alpha -\beta e^{-\kappa t}\right ) \right ]$ this is the damping law of classical Weyl correspondence of density operator in amplitude dessipative channel. The integration method within Weyl ordered product of operators and Wigner operator’s Wel ordering form ${\Delta } \left (\alpha ,\alpha ^{\ast } \right ) =\frac {1}{2}{{\begin {array}{l}:\\:\end {array}}}\delta \left (\alpha ^{\ast } -a^{\dag } \right ) \left (\alpha -a\right ){{\begin {array}{l}:\\:\end {array}}}$ is essential to our derivation.

For the density operator describing s−photon-subtracted squeezed chaotic light (PSSCL) we search for its thermo vacuum state (a pure state) in the real-fictitious space. We find that it reduces to a thermo vacuum state of squeezed chaotic light when s = 0, and to a thermo vacuum state of the optical negative binomial field when no squeezing. The new thermo vacuum state simplifies calculating photon number average, quantum fluctuation and Mandel’s Q parameter of PSSCL. Using the method of integration within ordered product (IWOP) of operators we also derive the normalization coefficient and explicitly analytical expressions of Wigner function for PSSCL.

We construct a new optical field, denoted as $\rho _{0}\equiv \left \vert \psi \right \rangle _{ll}\left \langle \psi \right \vert $ , $\left \vert \psi \right \rangle _{l}=\frac {\text {sech}^{l}\lambda }{\sqrt {l!} }b^{\dagger l}S_{2}\left \vert 00\right \rangle $ , by adding single-mode l-photon to a two-mode squeezed vacuum state, where $S_{2}=\exp \left [ \lambda \left (a^{\dagger }b^{\dagger }-ab\right ) \right ] $ is the two-mode squeezing operator. We find that its partial trace over b-mode will lead to a negative-binomial optical field in a-mode characteristic of l, which exhibits quantum entanglement. The photon number fluctuation of the new optical field both in a-mode and in b-mode is investigated. We employ the summation method within ordered product of operators to proceed our discussion.

Because the two coordinates of particle's guiding center of motion in uniform magnetic field do not commute, we think it is of necessity to study adiabatic invariant of electron motion in slowly varying magnetic field in the context of quantum mechanics. By constructing the entangled state representation which is the eigenvector of electron's coordinates and the squeezing mechanism of gyration radius, we directly reach the conclusion that the magnetic flux is adiabatic invariant in slowly varying magnetic field. We also compare this case with the adiabatic invariant of a pendulum whose string length is shortened very slowly.

We derive the evolution law of an initial two-mode squeezed vacuum state \( \text {sech}^{2}\lambda e^{a^{\dag }b^{\dagger }\tanh \lambda }\left \vert 00\right \rangle \left \langle 00\right \vert e^{ab\tanh \lambda }\) (a pure state) passing through an a-mode diffusion channel described by the master equation $$\frac{d\rho \left( t\right) }{dt}=-\kappa \left[ a^{\dagger}a\rho \left( t\right) -a^{\dagger}\rho \left( t\right) a-a\rho \left( t\right) a^{\dagger}+\rho \left( t\right) aa^{\dagger}\right] , $$ since the two-mode squeezed state is simultaneously an entangled state, the final state which emerges from this channel is a two-mode mixed state. Performing partial trace over the b-mode of ρ(t) yields a new chaotic field, \(\rho _{a}\left (t\right ) =\frac {\text {sech}^{2}\lambda }{1+\kappa t \text {sech}^{2}\lambda }:\exp \left [ \frac {- \text {sech}^{2}\lambda }{1+\kappa t\text {sech}^{2}\lambda }a^{\dagger }a \right ] :,\) which exhibits higher temperature and more photon numbers, showing the diffusion effect. Besides, measuring a-mode of ρ(t) to find n photons will result in the collapse of the two-mode system to a new Laguerre polynomial-weighted chaotic state in b-mode, which also exhibits entanglement.

By searching for Q- and P-ordering form of the coordinate-momentum intermediate representation |x⟩λ,νλ,ν⟨x| (formed by the eigenvector of λQ+νP with eigenvalue being x, Q,P=iħ), and noting that |x⟩λ,νλ,ν⟨x|=δ(x-λQ-νP) is just the Radon transformation of the Wigner operator, we find new relationship between a quantum state ψ’s tomogram and its wave function in coordinate (or momentum) representation, i.e. the tomogram can be split into two parts, one is ψq’s Gaussian integration transformation with the parameter ν and another is ψ∗p’s Gaussian integration transformation with the parameter λ. In this way, the quantum tomogram of number state is conveniently deduced. We also derive Radon transform of δ(p-P)δ(q-Q) and δ(q-Q)δ(p-P), which can be either viewed as P- and Q-ordered operator correspondence of the classical function δ(x-λq-νp).

We find a new relationship between Collins diffraction integration and quantum tomogram. We conclude that the tomogram of quantum state | ψ 〉 is just the module square of the Fresnel transform function of ψ ( x ), and this transform is just performed through the Collins formula, which opens up a new application area of Collins formula in calculating quantum tomogram of various quantum states.

In this paper we generalize Dirac’s phase operator \(\frac {1}{\sqrt {N}}a\) by defining a new phase operator \(\sqrt {\frac {a+\tilde {a}^{\dagger }}{a^{\dagger }+\tilde {a}}}\) in doubled Fock space, where \(\tilde {a}\) is a fictitious mode which annihilates the fictitious vacuum state \(\left \vert \tilde {0}\right \rangle \). It turns out that \(\sqrt {\frac {a+\tilde {a}^{\dagger }}{a^{\dagger }+\tilde {a}}}\) corresponds to a classical phase in the entangled state representation and is unitary. Remarkably, \(\left \langle \tilde {0}\right \vert \sqrt {\frac {a+\tilde {a}^{\dagger }}{a^{\dagger }+\tilde {a}}}\left \vert \tilde {0}\right \rangle \) is just the Paul’s phase operator whose antinormally ordered form is \({\vdots } \frac {1}{\sqrt {N}}a\vdots \). We also employ the method of integration within ordered product of operators to obtain the Fock representation of Paul’s phase operator, from which one can see how it deffers from Susskind-Glogower’s phase operator.