Quantum Mechanical Version for Bessel Beam＇s Propagation in ABCD Optical System

By introducing the entangled state representation and the two-mode Fresnel operator we provide quantum mechanical version for Bessel beam＇s classical propagation in ABCD optical system. This provides the opportunity of studying various classical Fresnel transformations in the context of quantum optics.     

Quantum optical ABCD theorem in two-mode case

By introducing the entangled Fresnel operator （EFO） this paper demonstrates that there exists ABCD theorem for two-mode entangled case in quantum optics. The canonical operator method as mapping of ray-transfer ABCD matrix is explicitly shown by EFO＇s normally ordered expansion through the coherent state representation and the technique of integration within an ordered product of operators.     

Optical Four-Wave Mixing Operator, Fresnel Operators and Three-Mode Entangled State Representation

We analyse the optical four-wave mixing operator S and relate it to the two-mode Fresnel operator. It is shown that the direct product of the two-mode Fresnel operator and the single-mode Fresnel operator has a natural representation on the basis of a three-mode entangled state, which is constructed by S and a beam splitter transform.     

A new theorem relating quantum tomogram to the Fresnel operator

According to Fan-Hu's formalism (Fan Hong-Yi and Hu Li-Yun 2009 Opt. Commun. bf282 3734) that the tomogram of quantum states can be considered as the module-square of the state wave function in the intermediate coordinate-momentum representation which is just the eigenvector of the Fresnel quadrature phase, we derive a new theorem for calculating quantum tomogram of density operator, i.e., the tomogram of a density operator ρ is equal to the marginal integration of the classical Weyl correspondence function of F⁺ρF, where F is the Fresnel operator. Applications of this theorem to evaluating the tomogram of optical chaotic field and squeezed chaotic optical field are presented.     

Radon transforms of the Wigner operator on hyperplanes

The generalization of tomographic maps to hyperplanes is considered. We find that the Radon transform of the Wigner operator in multi-dimensional phase space leads to a normally ordered operator in binomial distribution---a mixed-state density operator. Reconstruction of the Wigner operator is also feasible. The normally ordered form and the Weyl ordered form of the Wigner operator are used in our derivation. The operator quantum tomography theory is expressed in terms of some operator identities, with the merit of revealing the essence of the theory in a simple and concise way.     

Quantum mechanical version of the classical Liouville theorem

In terms of the coherent state evolution in phase space,we present a quantum mechanical version of the classical Liouville theorem.The evolution of the coherent state from |z>to|sz-rz*> corresponds to the motion from a point z(q,p) to another point sz-rz* with |s|2-|r|2=1.The evolution is governed by the so-called Fresnel operator U(s,r) that was recently proposed in quantum optics theory,which classically corresponds to the matrix optics law and the optical Fresnel transformation,and obeys group product rules.In other words,we can recapitulate the Liouville theorem in the context of quantum mechanics by virtue of coherent state evolution in phase space,which seems to be a combination of quantum statistics and quantum optics.     

General Wigner Transforms Studied by Virtue of Weyl Ordering of the Wigner Operator

By virtue of the property that Weyl ordering is invariant under similar transformations we show that the Weyl ordered form of the Wigner operator, a Dirac δ-operator function, brings much convenience for derivingmiscellaneous Wigner transforms. The operators which engender various transforms of the Wigner operator, can alsobe easily deduced by virtue of the Weyl ordering technique. The correspondence between the optical Wigner transformsand the squeezing transforms in quantum optics is investigated.     

New approach to Q-P （P-Q） ordering of quantum mechanical operators and its applications

In this paper, we introduce a new way to obtain the Q-P （P-Q） ordering of quantum mechanical operators, i.e., from the classical correspondence of Q-P （P-Q） ordered operators by replacing q and p with coordinate and momentum operators, respectively. Some operator identities are derived concisely. As for its applications, the single （two-） mode squeezed operators and Fresnel operator are examined. It is shown that the classical correspondence of Fresnel operator’s Q-P （P-Q） ordering is just the integration kernel of Fresnel transformation. In addition, a new photo-counting formula is constructed by the Q-P ordering of operators.     

Quantum Optics of Ultracold Molecular Fields

We apply concepts of quantum optical coherence to characterize the coherent generation of a molecular field from a quantum-degenerate atomic sample, and discuss the impact of the quantum statistics of the atoms on that field. For atoms initially in a BEC the resulting molecular field is to a good approximation coherent. This is in sharp contrast to the case of atoms in a normal Fermi gas, where we can made use of an analogy with the Tavis-Cummings model to show that the statistics of the resulting molecular field is similar to that of a single-mode chaotic light field. The BCS case interpolates between the two extremes, with an ＇incoherent＇ contribution from unpaired atoms superposed to a ＇coherent＇ contribution from atomic Cooper pairs. We also comment on the temporal fluctuations characteristic of the formation of molecular dimers from ultracold fermionic atoms.     

General Wigner Transforms Studied by Virtue of Weyl Ordering of the Wigner Operator

By virtue of the property that Weyl ordering is invariant under similar transformations we show that the Weyl ordered form of the Wigner operator, a Dirac δ-operator function, brings much convenience for deriving miscellaneous Wigner transforms. The operators which engender various transforms of the Wigner operator, can also be easily deduced by virtue of the Weyl ordering technique. The correspondence between the optical Wigner transforms and the squeezing transforms in quantum optics is investigated.     

Application of Coordinate-Momentum Intermediate Representation in Solving Eigenstate Problem of a Coupled Oscillator System

We study the eigenstate problem of a kind of coupled oscillators in the new quantum mechanical representation ｜q,μ,υ〉, which is defined as the eigenvector of the operator （μQ ＋ υP）, whereμ and υ are two real parameters. We also use the U operator transformation method to deal with the same problem. We obtain the normally ordered product expressions of U operator and eigenvector. It is shown that the ground state of system Hamiltonian is a squeezed state.     

Application of Coordinate-Momentum Intermediate Representation in Solving Eigenstate Problem of a Coupled Oscillator System

We study the eigenstate problem of a kind of coupled oscillators in the new quantum mechanical represen tation ｜q, μ, v〉, which is defined as the eigenvector of the operator (μQ vP), where μ and v are two realparameters. We also use the U operator transformation method to deal with the same problem. We obtain the normally ordered product expressions of U operator and eigenvector. It is shown that the ground state of system Hamiltonian is a squeezed state.     

Quantum Geometrical Phases and Mesoscopic Persistent Current

We critically examine the concepts of quantum geometrical phases which have been widely applied to calculate mesoscopic persistent currents in the current literature.We point out that the method used in these calculations is essentially in conflict with the basic concepts of quantum geometrical phases.We give an example calculation of mesoscopic ring Aharonov-Anandan,Aharonov-Casher phases and persistent current to show some of the misconceptions involved in the current literature.     

Supersymmetric Quantum Mechanics and SUSY Dependent SU（2） Symmetry for a Spin-1/2 Charged Particle in a Magnetic Field

We find that in a supersymmetric quantum mechanics （SUSY QM） system, in addition to supersymmetric algebra, an associated SU（2） algebra can be obtained by using semiunitary （SUT） operator and projection operator, and the relevant constants of motion can be constructed. Two typical quantum systems are investigated as examples to demonstrate the above finding. The first example is the quantum system of a nonrelativistic charged particle moving in x-y plane and coupled to a magnetic field along z-axis. The second example is provided with the Dirac particle in a magnetic field. Similarly there exists an SUτ（2） SUσ（2） symmetry in the context of the relativistic Pauli Hamilt onian squared. We show that there exists also an SU（2） symmetry associated with the supersvmmetrv of the Dirac particle.     

Optical scaled Fresnel--Fourier transform obtained via intermediate coordinate-- momentum representation

Using the intermediate coordinate--momentum representation |x>_s,r, we introduce a new Hadamard transform. It is found that the operator U corresponding to this transform can be considered as a combination of the Fresnel operator F(r,s) and the Fourier transform operator F by decomposing U. We also find that the matrix element _s,r< x| U|f> just corresponds to an optical scaled Fresnel--Fourier transform.     

Time Evolution Caused by Hamiltonian Composed of Quadratic Combination of Canonical Operators and Time-Dependent Two-Mode Fresnel Operator

We show that the time-dependent two-mode Fresnel operator is just the time-evolutional unitary operator governed by the Hamiltonian composed of quadratic combination of canonical operators in the way of exhibiting SU（1,1） algebra. This is an approach for obtaining the time-dependent Hamiltonian from the preassigned time evolution in classical phase space, an approach which is in contrast to Lewis-Riesenfeld＇s invariant operator theory of treating timedependent harmonic oscillators.     

Time Evolution Caused by Hamiltonian Composed of Quadratic Combination of Canonical Operators and Time-Dependent Two-Mode Fresnel Operator

We show that the time-dependent two-mode Fresnel operator is just the time-evolutional unitary operator governed by the Hamiltonian composed of quadratic combination of canonical operators in the way of exhibiting SU(1,1)algebra. This is an approach for obtaining the time-dependent Hamiltonian from the preassigned time evolution in classical phase space, an approach which is in contrast to Lewis-Riesenfeld‘s invariant operator theory of treating time-dependent harmonic oscillators.     

Simple Schemes of CQED for Realizing Swap, Entangling and CNOT Quantum Gates

We consider two cavities which are spatially separated and connected by an optical fibre. There are multi two-level atoms in each of the cavities. The atoms resonantly interact with the cavity fields but there is no direct interaction between the atoms. We show that perfect swap and entangling quantum gates can be realised between the two atoms clusters if modes of the electromagnetic field in the cavities and fibre are initially not excited. Compared with the single atom scheme, we find that the multi-atom scheme can speed up the quantum gates by a factor √N where N is number of the atoms in each of the cavities. We also consider the case where two two-level atoms in distant cavities that are coupled by an optical fibre. We find if both of the atoms interact resonantly with the fields, a highly reliable CNOT gate can be achieved within much less operation time than that of the non- resonant case. The sensibility of these gates to various parameters contained in the models under consideration is also investigated.     

New generating function formulae of even- and odd-Hermite polynomials obtained and applied in the context of quantum optics

By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials which will be useful in constructing new optical field states. We then show that the squeezed state and photon-added squeezed state can be expressed by even- and odd-Hermite polynomials.     

A generalized Collins formula derived by virtue of the displacement-squeezing related squeezed coherent state representation

Based on the displacement-squeezing related squeezed coherent state representation ≤ft\vert z\right\rangle _{g} and using the technique of integration within an ordered product of operators, this paper finds a generalized Fresnel operator, whose matrix element in the coordinate representation leads to a generalized Collins formula (Huygens--Fresnel integration transformation describing optical diffraction). The generalized Fresnel operator is derived by a quantum mechanical mapping from z to sz-rz^{\ast } in the % ≤ft\vert z\right\rangle _{g} representation, while ≤ft\vert z\right\rangle _{g} in phase space is graphically denoted by an ellipse.