Integration Within An Ordered Product Technique Research Articles

Research on mesoscopic inductance–capacitance coupling circuit employing an unitary operator and the IWOP technique

Employing a special unitary operator and the virtue of the technique of integration within an ordered product (IWOP), we research on the quantum effects for an inductance–capacitance coupling double L–C circuit. It is found that it is convenient to research on the mesoscopic circuit with such an operator. First, eliminating the coupling term of Hamiltonian becomes easier than other methods. Second, the system’s ground state can be easily generalized into a two-mode squeezed state. Thus, the quantum fluctuations of the circuit and the variances of current in each loop are calculated, the charge-current uncertainty relation is revealed. As a result, both the uncertainty relation and fluctuation of the current in each loop become larger when [Formula: see text] is larger or [Formula: see text] is smaller.

New operator identities with regard to the two-variable Hermite polynomial by virtue of entangled state representation

By virtue of the entangled state representation we concisely derive some new operator identities with regard to the two-variable Hermite polynomial (TVHP). By them and the technique of integration within an ordered product (IWOP) of operators we further derive new generating function formulas of the TVHP. They are useful in quantum optical theoretical calculations. It is seen from this work that by combining the IWOP technique and quantum mechanical representations one can derive some new integration formulas even without really performing the integration.

New Route to Deducing Integration Formulas by Virtue of the IWOP Technique

We point out a new route to deducing integration formulas, i.e., using the technique of integration within an ordered product (IWOP) of operators we derive some new integration formulas, which seems concise. As a by-product, some new operator identities also appear.

FINDING AND DISENTANGLING COMPLICATED UNITARY OPERATORS BY VIRTUE OF THE INTEGRATION OVER DIRAC'S KET–BRA OPERATORS

Usually complicated unitary operators in exponential form are hard to handle and the transformation stated by them are not physically clear until these operators are disentangled. By virtue of the technique of integration within an ordered product (IWOP) of operators, we can disentangle some complicated unitary operators and then reveal their physical role. Many new unitary operators can be found after new quantum mechanical representations are constructed by virtue of the IWOP technique. The unitary operators for permutation, Hilbert transform, Householder transform, and Hardmad transform can also be introduced. The IWOP technique thus bridges this mathematical gap between classical mechanics and quantum mechanics and provides a new route connecting classical transformations to quantum mechanical unitary operators. In this way, the symbolic method exhibits further beauty and elegance and can be increasingly used for developing various fields of quantum physics.

The q-analogues of two-mode squeezed states constructed by virtue of the IWOP technique

The q-analogues of two-mode squeezed states are introduced by virtue of deformation quantization methods and the technique of integration within an ordered product (IWOP) of operators. Some new completeness relations about these squeezed states composed of the bra and ket which are not mutually Hermitian conjugates are obtained. Furthermore, the antibunching effects of the two-mode squeezed vacuum state S′2(r) |00〉 are investigated. It is found that, in different ranges of the squeezed parameter r, both modes of the state exhibit the antibunching effects and the two modes of the state are always nonclassical correlation.

Newton–Leibniz integration for ket–bra operators (III)—Application in fermionic quantum statistics

The Newton–Leibniz integration over Dirac’s ket–bra operators introduced in Ref. [Hong-yi Fan, Hai-liang Lu, Yue Fan, Ann. Phys. 321 (2006) 480–494] is generalized to Newton–Leibniz–Berezin integration over fermionic ket–bra projection operators, the corresponding technique of integration within an ordered product (IWOP) of Fermi operators is proposed which is then used to develop fermionic quantum statistics. The generalized partition function formula of multi-mode quadratic interacting fermion is derived via the fermionic coherent state representation and the IWOP technique. The two-mode fermionic squeezing operators and their group property studied by their fermionic coherent state representation as well as fermionic permutation operator are also deduced in this way. Thus Dirac’s symbolic method for Fermi system can also be developed, which exhibits Bose-Fermi supersymmetry in this aspect.

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.

ENTANGLED STATES, SQUEEZED STATES GAINED VIA THE ROUTE OF DEVELOPING DIRAC'S SYMBOLIC METHOD AND THEIR APPLICATIONS

Via the route of developing Dirac's symbolic method by virtue of the technique of integration within an ordered product (IWOP) of operators we derive various entangled state representations, squeezed states, and operator ordering formulae. We show how the entangled states can be applied to many aspects of quantum optics theory. The applications of the entangled states in the interdisciplinary study of quantum optics and Fourier optics are also introduced. All these discussions exhibit the power of Dirac's symbolic method with the aid of the IWOP technique.

Operator ordering in quantum optics theory and the development of Dirac s symbolic method

We present a general unified approach for arranging quantum operators of optical fields into ordered products (normal ordering, antinormal ordering, Weyl ordering (or symmetric ordering)) by fashioning Dirac's symbolic method and representation theory. We propose the technique of integration within an ordered product (IWOP) of operators to realize our goal. The IWOP makes Dirac's representation theory and the symbolic method more transparent and consequently more easily understood. The beauty of Dirac's symbolic method is further revealed. Various applications of the IWOP technique, such as in developing the entangled state representation theory, nonlinear coherent state theory, Wigner function theory, etc, are presented.

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.

Squeezing in the triatomic linear molecule model revealed by virtue of the IWOP technique

For a typical triatomic linear molecule with different atomic masses one finds a unitary operator which is closely related to squeezing transformations caused by frequency jumps. The coordinate representation of the unitary operator is identified, which provides an alternative approach for calculating the wavefunction of internal motion of the molecule. The author also shows that the unitary operator is equivalent to a two-mode squeezing-rotating combination operator multiplied from the right by two single-mode squeezing operators. The discussion is mainly based on the technique of integration within an ordered product (IWOP) of operators.

Antinormally ordering some multimode exponential operators by virtue of the IWOP technique

The author shows that the technique of integration within an ordered product (IWOP) provides one with a very simple approach to deriving the normal product form of some multimode exponential operators, which greatly simplifies the calculations of normalising some state vectors in Hilbert space.

Normally ordering some multimode exponential operators by virtue of the IWOP technique

The author shows that the technique of integration within an ordered product (IWOP) provides one with a very simple approach to deriving the normal product form of some multimode exponential operators, which greatly simplifies the calculations of normalising some state vectors in Hilbert space.