We consider the quantum mechanical SU（2） transformation e^2λJzJ±e^-2λJz = e^±2λJ±as if the meaning of squeezing with e^±2λ being squeezing parameter. By studying SU（2） operators （J±,...We consider the quantum mechanical SU（2） transformation e^2λJzJ±e^-2λJz = e^±2λJ±as if the meaning of squeezing with e^±2λ being squeezing parameter. By studying SU（2） operators （J±, Jz） from the point of view of squeezing we find that （J±,Jz） can also be realized in terms of 3-mode bosonic operators. Employing this realization, we find the natural representation （the eigenvectors of J＋ or J-） of the 3-mode squeezing operator e^2λJz. The idea of considering quantum SU（2） transformation as if squeezing is liable for us to obtain the new bosonic operator realization of SU（2） and new squeezing operators.展开更多
Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting proper...Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.展开更多
The effect of the field–field interaction on a cavity containing two qubit(TQ)interacting with a two mode of electromagnetic field as parametric amplifier type is investigated.After performing an appropriate transfor...The effect of the field–field interaction on a cavity containing two qubit(TQ)interacting with a two mode of electromagnetic field as parametric amplifier type is investigated.After performing an appropriate transformation,the constants of motion are calculated.Using the Schrödinger differential equation a system of differential equations was obtained,and the general solution was obtained in the case of exact resonance.Some statistical quantities were calculated and discussed in detail to describe the features of this system.The collapses and revivals phenomena have been discussed in details.The Shannon information entropy has been applied for measuring the degree of entanglement(DE)between the qubits and the electromagnetic field.The normal squeezing for some values of the parameter of the field–field interaction is studied.The results showed that the collapses disappeared after the field–field terms were added and the maximum values of normal squeezing decrease when increasing of the field–field interaction parameter.While the revivals and amplitudes of the oscillations increase when the parameter of the field–field interaction increases.Degree of entanglement is partially more entangled with increasing of the field-field interaction parameter.The relationship between revivals,collapses and the degree of entanglement(Shannon information entropy)was monitored and discussed in the presence and absence of the field–field interaction.展开更多
This paper presents sufficient and necessary conditions for the propagator controllability of a class of infinite-dimensional quantum systems with SU(1,1)dynamical symmetry through the isomorphic mapping to the non-un...This paper presents sufficient and necessary conditions for the propagator controllability of a class of infinite-dimensional quantum systems with SU(1,1)dynamical symmetry through the isomorphic mapping to the non-unitary representation of SU(1,1).The authors prove that the elliptic condition of the total Hamiltonian is both necessary and sufficient for the controllability and strong controllability.The obtained results can be also extended to control systems with SO(2,1)dynamical symmetry.展开更多
基金supported by the National Natural Science Foundation of China(Grant Nos.11175113 and 11275123)the Key Project of Natural Science Fund of Anhui Province,China(Grant No.KJ2013A261)
文摘We consider the quantum mechanical SU（2） transformation e^2λJzJ±e^-2λJz = e^±2λJ±as if the meaning of squeezing with e^±2λ being squeezing parameter. By studying SU（2） operators （J±, Jz） from the point of view of squeezing we find that （J±,Jz） can also be realized in terms of 3-mode bosonic operators. Employing this realization, we find the natural representation （the eigenvectors of J＋ or J-） of the 3-mode squeezing operator e^2λJz. The idea of considering quantum SU（2） transformation as if squeezing is liable for us to obtain the new bosonic operator realization of SU（2） and new squeezing operators.
文摘Starting from general Jacobi polynomials we derive for the Ul-traspherical polynomials as their special case a set of related polynomials which can be extended to an orthogonal set of functions with interesting properties. It leads to an alternative definition of the Ultraspherical polynomials by a fixed integral operator in application to powers of the variable u in an analogous way as it is possible for Hermite polynomials. From this follows a generating function which is apparently known only for the Legendre and Chebyshev polynomials as their special case. Furthermore, we show that the Ultraspherical polynomials form a realization of the SU(1,1) Lie algebra with lowering and raising operators which we explicitly determine. By reordering of multiplication and differentiation operators we derive new operator identities for the whole set of Jacobi polynomials which may be applied to arbitrary functions and provide then function identities. In this way we derive a new “convolution identity” for Jacobi polynomials and compare it with a known convolution identity of different structure for Gegenbauer polynomials. In short form we establish the connection of Jacobi polynomials and their related orthonormalized functions to the eigensolution of the Schrödinger equation to Pöschl-Teller potentials.
基金the University of Jeddah,Saudi Arabia,under Grant No.UJ-02-082-DR.
文摘The effect of the field–field interaction on a cavity containing two qubit(TQ)interacting with a two mode of electromagnetic field as parametric amplifier type is investigated.After performing an appropriate transformation,the constants of motion are calculated.Using the Schrödinger differential equation a system of differential equations was obtained,and the general solution was obtained in the case of exact resonance.Some statistical quantities were calculated and discussed in detail to describe the features of this system.The collapses and revivals phenomena have been discussed in details.The Shannon information entropy has been applied for measuring the degree of entanglement(DE)between the qubits and the electromagnetic field.The normal squeezing for some values of the parameter of the field–field interaction is studied.The results showed that the collapses disappeared after the field–field terms were added and the maximum values of normal squeezing decrease when increasing of the field–field interaction parameter.While the revivals and amplitudes of the oscillations increase when the parameter of the field–field interaction increases.Degree of entanglement is partially more entangled with increasing of the field-field interaction parameter.The relationship between revivals,collapses and the degree of entanglement(Shannon information entropy)was monitored and discussed in the presence and absence of the field–field interaction.
基金supported by the National Natural Science Foundation of China under Grant Nos.61803357,61833010,61773232,61622306 and 11674194the National Key R&D Program of China under Grant Nos.2018YFA0306703 and 2017YFA0304304+1 种基金the Tsinghua University Initiative Scientific Research Programthe Tsinghua National Laboratory for Information Science and Technology Cross-discipline Foundation。
文摘This paper presents sufficient and necessary conditions for the propagator controllability of a class of infinite-dimensional quantum systems with SU(1,1)dynamical symmetry through the isomorphic mapping to the non-unitary representation of SU(1,1).The authors prove that the elliptic condition of the total Hamiltonian is both necessary and sufficient for the controllability and strong controllability.The obtained results can be also extended to control systems with SO(2,1)dynamical symmetry.
