Based on the cascade two-photon laser dynamic equation derived with the technique of quantum Langevin operators with the considerations of coherently prepared three-level atoms and the classical field injected into th...Based on the cascade two-photon laser dynamic equation derived with the technique of quantum Langevin operators with the considerations of coherently prepared three-level atoms and the classical field injected into the cavity, we numerically study the effects of atomic coherence and classical field on the chaotic dynamics of a two-photon laser. Lyapunov exponent and bifurcation diagram calculations show that the Lorenz chaos and hyperchaos can be induced or inhibited by the atomic coherence and the classical field via crisis or Hopf bifurcations.展开更多
A scheme is proposed to generate the W-type entangled coherent states of three-cavity field. The scheme is based on the resonant atom-field interaction, thus the interaction time between the atom and the cavity is gre...A scheme is proposed to generate the W-type entangled coherent states of three-cavity field. The scheme is based on the resonant atom-field interaction, thus the interaction time between the atom and the cavity is greatly reduced, which is important in view of decoherence. Furthermore, the scheme does not need accurate adjustment of the interaction time.展开更多
We have studied the dynamics and transfer of the entanglement of the two identical atoms simultaneously interacting with vacuum field by employing the dressed-state representation. The two atoms are driven by classica...We have studied the dynamics and transfer of the entanglement of the two identical atoms simultaneously interacting with vacuum field by employing the dressed-state representation. The two atoms are driven by classical fields. The influence of the initial entanglement degree of two atoms, the coupling strength between the atom and the classical field and the detuning between the atomic transition frequency and the frequency of classical field on the entanglement and atomic linear entropy is discussed. The initial entanglement of the two atoms can be transferred into the entanglement between the atom and cavity field when the dissipation is neglected. The maximally entangled state between the atoms and cavity field can be obtained under some certain conditions. The time of disentanglement of two atoms can be controlled and manipulated by adjusting the detuning and classical driving fields. Moreover, the larger the cavity decay rate is, the more quickly the entanglement of the two atoms decays.展开更多
We investigate a two=level atom interacting with a quantized cavity field and a classical driving field in the presence of phase decoherence and find that a stationary quantum discord can arise in the interaction of t...We investigate a two=level atom interacting with a quantized cavity field and a classical driving field in the presence of phase decoherence and find that a stationary quantum discord can arise in the interaction of the atom and cavity field as the time turns to infinity. We also find that the stationary quantum discord can be increased by applying a classical driving field. Furthermore, we explore the quantum discord dynamics of two identical non=interacting two-level atoms independently interacting with a quantized cavity field and a classical driving field in the presence of phase decoherence. Results show that the quantum discord between two atoms is more robust than entanglement under phase decoherence and the classical driving field can help to improve the amount of quantum discord of the two atoms.展开更多
We investigate the dynamics of entanglement through negativity and witness operators in a system of four noninteracting qubits driven by a classical phase noisy laser characterized by a classical random external field...We investigate the dynamics of entanglement through negativity and witness operators in a system of four noninteracting qubits driven by a classical phase noisy laser characterized by a classical random external field(CREF).The qubits are initially prepared in the GHZ-type and W-type states and interact with the CREF in two different qubit-field configurations,namely,common environment and independent environments in which the cases of equal and different field phase probabilities are distinguished.We find that entanglement exhibits different decaying behavior,depending on the input states of the qubits,the qubit-field coupling configuration,and field phase probabilities.On the one hand,we demonstrate that the coupling of the qubits in a common environment is an alternative and more efficient strategy to completely shield the system from the detrimental impacts of the decoherence process induced by a CREF,independent of the input state and the field phase probabilities considered.Also,we show that GHZ-type states have strong dynamics under CREF as compared to W-type states.On the other hand,we demonstrate that in the model investigated the system robustness’s can be greatly improved by increasing the number of qubits constituting the system.展开更多
A systematic method is developed to studY the classical motion of a mass point in gravitational gauge field. First, by using Mathematica, a spherical symmetric solution of the field equation of gravitational gauge fie...A systematic method is developed to studY the classical motion of a mass point in gravitational gauge field. First, by using Mathematica, a spherical symmetric solution of the field equation of gravitational gauge field is obtained, which is just the traditional Schwarzschild solution. Combining the principle of gauge covariance and Newton＇s second law of motion, the equation of motion of a mass point in gravitational field is deduced. Based on the spherical symmetric solution of the field equation and the equation of motion of a mass point in gravitational field, we can discuss classical tests of gauge theory of gravity, including the deflection of light by the sun, the precession of the perihelia of the orbits of the inner planets and the time delay of radar echoes passing the sun. It is found that the theoretical predictions of these classical tests given by gauge theory of gravity are completely the same as those given by general relativity.展开更多
The exact dynamics of an open quantum system consisting of one qubit driven by a classical driving field is investigated. Our attention is focused on the influences of single-and two-photon excitations on the dynamics...The exact dynamics of an open quantum system consisting of one qubit driven by a classical driving field is investigated. Our attention is focused on the influences of single-and two-photon excitations on the dynamics of quantum coherence and quantum entanglement. It is shown that the atomic coherence can be improved or even maintained by the classical driving field, the non-Markovian effect, and the atom-reservoir detuning. The interconversion between the atomic coherence and the atom-reservoir entanglement exists and can be controlled by the appropriate conditions. The conservation of coherence for different partitions is explored, and the dynamics of a system with two-photon excitations is different from the case of single-photon excitation.展开更多
In this work, we examine the geometric character of the field equations of general relativity and propose to formulate relativistic field equations in terms of the Riemann curvature tensor. The resulted relativistic f...In this work, we examine the geometric character of the field equations of general relativity and propose to formulate relativistic field equations in terms of the Riemann curvature tensor. The resulted relativistic field equations are also integrated into the general framework that we have presented in our previous works that all known classical fields can be expressed in the same dynamical form. We also discuss a possibility to reformulate the field equations of general relativity so that the Ricci curvature tensor and the energy-momentum tensor can appear symmetrically in the field equations without violating the conservation law stated by the covariant derivative.展开更多
We investigate the quantum coherence and quantum entanglement dynamics of a classical driven single atom coupled to a single-mode cavity. It is shown that the transformation between the atomic coherence and the atom-f...We investigate the quantum coherence and quantum entanglement dynamics of a classical driven single atom coupled to a single-mode cavity. It is shown that the transformation between the atomic coherence and the atom-field entanglement exists, and can be improved by adjusting the classical driving field. The joint evolution of two identical single-body systems is also studied. The results show the quantum coherence transfers among composite subsystems, and the coherence conservation of composite subsystems is obtained. Moreover, the classical driving field can be used to suppress the decay of the coherence and entanglement, owing to considering the leaky cavity. The non-Markovian dynamics of the system is also discussed finally.展开更多
By using a two-dimensional Monte-Carlo classical ensemble method, we investigate the double ionization（DI） process of the CS_2 molecule with different bond lengths in an 800-nm intense laser field. The double ioniza...By using a two-dimensional Monte-Carlo classical ensemble method, we investigate the double ionization（DI） process of the CS_2 molecule with different bond lengths in an 800-nm intense laser field. The double ionization probability presents a ＂knee＂ structure with equilibrium internuclear distance R = 2.9245 a.u.（a.u. is short for atomic unit）. As the bond length of CS increases, the DI probability is enhanced and the ＂knee＂ structure becomes less obvious. In addition,the momentum distribution of double ionized electrons is also investigated, which shows the momentum mostly distributed in the first and third quadrants with equilibrium internuclear distance R = 2.9245 a.u. As the bond length of CS increases,the electron momentum becomes evenly distributed in the four quadrants. Furthermore, the energy distributions and the corresponding trajectories of the double-ionized electrons versus time are also demonstrated, which show that the bond length of CS in the CS_2 molecule plays a key role in the DI process.展开更多
A classical field theory for a Schrodinger equation with a non-Hermitian Hamiltonian describing a particle with position-dependent mass has been recently advanced by Nobre and Rego-Monteiro(NR)[Phys.Rev.A 88(2013)0321...A classical field theory for a Schrodinger equation with a non-Hermitian Hamiltonian describing a particle with position-dependent mass has been recently advanced by Nobre and Rego-Monteiro(NR)[Phys.Rev.A 88(2013)032105].This field theory is based on a variational principle involving the wavefunction Ψ(x,t) and an auxiliary fieldΦ{x,t).It is here shown that the relation between the dynamics of the auxiliary field Φ(x,t) and that of the original wavefunction Ψ(x,t) is deeper than suggested by the NR approach.Indeed,we formulate a variational principle for the aforementioned Schrodinger equation which is based solely on the wavefunction Ψ(x,t).A continuity equation for an appropriately defined probability density,and the concomitant preservation of the norm,follows from this variational principle via Noether's theorem.Moreover,the norm-conservation law obtained by NR is reinterpreted as tie preservation of the inner product between pairs of solutions of the variable mass Schrodinger equation.展开更多
We propose two physical schemes, which can teleport unknown atomic entangled states from user A （Alice） to user B （Bob） via GHZ class states as quantum channel The two schemes are both based on cavity QED techniqu...We propose two physical schemes, which can teleport unknown atomic entangled states from user A （Alice） to user B （Bob） via GHZ class states as quantum channel The two schemes are both based on cavity QED techniques. In the two schemes, teleportation and distillation procedures can be realized simultaneously. The second teleportation scheme is more advantageous than the first one.展开更多
We investigate the entanglement dynamics of a quantum system consisting of two two-level atoms in a cavity with classical driving fields in the presence of white noise. The cavity is initially prepared in the vacuum s...We investigate the entanglement dynamics of a quantum system consisting of two two-level atoms in a cavity with classical driving fields in the presence of white noise. The cavity is initially prepared in the vacuum state. Generally, the entanglement of two atoms decreases with the intensity of the thermal fields and the coupling strength of the two-level atoms to the thermal fields. However, we find that the entanglement of the quantum system can be enhanced by adjusting the frequency and the strength of the classical driving fields in the presence of white noise.展开更多
We generalize several well known quantum equations to a Tsallis’ q-scenario, and provide a quantum version of some classical fields associated with them in the recent literature. We refer to the q-Schro¨dinger, ...We generalize several well known quantum equations to a Tsallis’ q-scenario, and provide a quantum version of some classical fields associated with them in the recent literature. We refer to the q-Schro¨dinger, q-KleinGordon, q-Dirac, and q-Proca equations advanced in, respectively, Phys. Rev. Lett. 106, 140601(2011), EPL 118,61004(2017) and references therein. We also introduce here equations corresponding to q-Yang-Mills fields, both in the Abelian and non-Abelian instances. We show how to define the q-quantum field theories corresponding to the above equations, introduce the pertinent actions, and obtain equations of motion via the minimum action principle.These q-fields are meaningful at very high energies(Te V scale) for q = 1.15, high energies(Ge V scale) for q = 1.001,and low energies(Me V scale) for q =1.000001 [Nucl. Phys. A 955(2016) 16 and references therein].(See the ALICE experiment at the LHC). Surprisingly enough, these q-fields are simultaneously q-exponential functions of the usual linear fields’ logarithms.展开更多
Let G be a classical group over an arbitrary field F,acting on an n-dimensional vector space V=V(n,F)over a field F.In this paper,we classify the maximal subgroups of G,which normalizes a solvable subgroup N of GL(L,F...Let G be a classical group over an arbitrary field F,acting on an n-dimensional vector space V=V(n,F)over a field F.In this paper,we classify the maximal subgroups of G,which normalizes a solvable subgroup N of GL(L,F)not lying in F^(*)1_(V).展开更多
Let G be a classical group over an arbitrary field F,acting on an n-dimensional F-space V=V(n,F).All those maximal subgroups of G are classified each of which normalizes a solvable subgroup N of GL(V/F)not lying in F^...Let G be a classical group over an arbitrary field F,acting on an n-dimensional F-space V=V(n,F).All those maximal subgroups of G are classified each of which normalizes a solvable subgroup N of GL(V/F)not lying in F^(*)1v.展开更多
We study the approaches to two-dimensional integrable field theories via a six-dimensional(6 D) holomorphic Chern-Simons theory defined on twistor space. Under symmetry reduction, it reduces to a 4 D Chern-Simons theo...We study the approaches to two-dimensional integrable field theories via a six-dimensional(6 D) holomorphic Chern-Simons theory defined on twistor space. Under symmetry reduction, it reduces to a 4 D Chern-Simons theory, while under solving along fibres it leads to a four-dimensional(4 D) integrable theory, the anti-self-dual Yang-Mills or its generalizations. From both 4 D theories, various two-dimensional integrable field theories can be obtained. In this work, we try to investigate several twodimensional integrable deformations in this framework. We find that the λ-deformation, the rational η-deformation, and the generalized λ-deformation can not be realized from the 4 D integrable model approach, even though they could be obtained from the 4 D Chern-Simons theory. The obstacle stems from the incompatibility between the symmetry reduction and the boundary conditions. Nevertheless, we show that a coupled theory of the λ-deformation and the η-deformation in the trigonometric description could be obtained from the 6 D theory in both ways, by considering the case that(3, 0)-form in the 6 D theory is allowed to have zeros.展开更多
基金The project partially supported by the Natural Science Foundation of Jiangsu Province of China under Grant No. BK2005062
文摘Based on the cascade two-photon laser dynamic equation derived with the technique of quantum Langevin operators with the considerations of coherently prepared three-level atoms and the classical field injected into the cavity, we numerically study the effects of atomic coherence and classical field on the chaotic dynamics of a two-photon laser. Lyapunov exponent and bifurcation diagram calculations show that the Lorenz chaos and hyperchaos can be induced or inhibited by the atomic coherence and the classical field via crisis or Hopf bifurcations.
基金The project supported by the Natural Science Foundation of Education Committee of Fujian Province of China under Grant No. JB03047.
文摘A scheme is proposed to generate the W-type entangled coherent states of three-cavity field. The scheme is based on the resonant atom-field interaction, thus the interaction time between the atom and the cavity is greatly reduced, which is important in view of decoherence. Furthermore, the scheme does not need accurate adjustment of the interaction time.
基金Supported by National Natural Science Foundation of China under Grant Nos.11247213,61368002,11304010,11264030,61168001China Postdoctoral Science Foundation under Grant No.2013M531558+4 种基金Jiangxi Postdoctoral Research Project under Grant No.2013KY33the Natural Science Foundation of Jiangxi Province under Grant No.20142BAB217001the Foundation for Young Scientists of Jiangxi Province(Jinggang Star)under Grant No.20122BCB23002the Research Foundation of the Education Department of Jiangxi Province under Grant Nos.GJJ13051,GJJ13057the Graduate Innovation Special Fund of Nanchang University under Grant No.cx2015137
文摘We have studied the dynamics and transfer of the entanglement of the two identical atoms simultaneously interacting with vacuum field by employing the dressed-state representation. The two atoms are driven by classical fields. The influence of the initial entanglement degree of two atoms, the coupling strength between the atom and the classical field and the detuning between the atomic transition frequency and the frequency of classical field on the entanglement and atomic linear entropy is discussed. The initial entanglement of the two atoms can be transferred into the entanglement between the atom and cavity field when the dissipation is neglected. The maximally entangled state between the atoms and cavity field can be obtained under some certain conditions. The time of disentanglement of two atoms can be controlled and manipulated by adjusting the detuning and classical driving fields. Moreover, the larger the cavity decay rate is, the more quickly the entanglement of the two atoms decays.
基金Project supported by National Natural Science Foundation of China (Grant No. 10774143)
文摘We investigate a two=level atom interacting with a quantized cavity field and a classical driving field in the presence of phase decoherence and find that a stationary quantum discord can arise in the interaction of the atom and cavity field as the time turns to infinity. We also find that the stationary quantum discord can be increased by applying a classical driving field. Furthermore, we explore the quantum discord dynamics of two identical non=interacting two-level atoms independently interacting with a quantized cavity field and a classical driving field in the presence of phase decoherence. Results show that the quantum discord between two atoms is more robust than entanglement under phase decoherence and the classical driving field can help to improve the amount of quantum discord of the two atoms.
文摘We investigate the dynamics of entanglement through negativity and witness operators in a system of four noninteracting qubits driven by a classical phase noisy laser characterized by a classical random external field(CREF).The qubits are initially prepared in the GHZ-type and W-type states and interact with the CREF in two different qubit-field configurations,namely,common environment and independent environments in which the cases of equal and different field phase probabilities are distinguished.We find that entanglement exhibits different decaying behavior,depending on the input states of the qubits,the qubit-field coupling configuration,and field phase probabilities.On the one hand,we demonstrate that the coupling of the qubits in a common environment is an alternative and more efficient strategy to completely shield the system from the detrimental impacts of the decoherence process induced by a CREF,independent of the input state and the field phase probabilities considered.Also,we show that GHZ-type states have strong dynamics under CREF as compared to W-type states.On the other hand,we demonstrate that in the model investigated the system robustness’s can be greatly improved by increasing the number of qubits constituting the system.
文摘A systematic method is developed to studY the classical motion of a mass point in gravitational gauge field. First, by using Mathematica, a spherical symmetric solution of the field equation of gravitational gauge field is obtained, which is just the traditional Schwarzschild solution. Combining the principle of gauge covariance and Newton＇s second law of motion, the equation of motion of a mass point in gravitational field is deduced. Based on the spherical symmetric solution of the field equation and the equation of motion of a mass point in gravitational field, we can discuss classical tests of gauge theory of gravity, including the deflection of light by the sun, the precession of the perihelia of the orbits of the inner planets and the time delay of radar echoes passing the sun. It is found that the theoretical predictions of these classical tests given by gauge theory of gravity are completely the same as those given by general relativity.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.61675115,11204156,11574178,11304179,and 11647172)the Science and Technology Plan Projects of Shandong University,China(Grant No.J16LJ52)
文摘The exact dynamics of an open quantum system consisting of one qubit driven by a classical driving field is investigated. Our attention is focused on the influences of single-and two-photon excitations on the dynamics of quantum coherence and quantum entanglement. It is shown that the atomic coherence can be improved or even maintained by the classical driving field, the non-Markovian effect, and the atom-reservoir detuning. The interconversion between the atomic coherence and the atom-reservoir entanglement exists and can be controlled by the appropriate conditions. The conservation of coherence for different partitions is explored, and the dynamics of a system with two-photon excitations is different from the case of single-photon excitation.
文摘In this work, we examine the geometric character of the field equations of general relativity and propose to formulate relativistic field equations in terms of the Riemann curvature tensor. The resulted relativistic field equations are also integrated into the general framework that we have presented in our previous works that all known classical fields can be expressed in the same dynamical form. We also discuss a possibility to reformulate the field equations of general relativity so that the Ricci curvature tensor and the energy-momentum tensor can appear symmetrically in the field equations without violating the conservation law stated by the covariant derivative.
基金supported by the National Natural Science Foundation of China(Grant Nos.61675115,11204156,11574178,and 11304179)the Science and Technology Plan Projects of Shandong University,China(Grant No.J16LJ52)the Natural Science Foundation of Shandong Province,China(Grant No.ZR2016AP09)
文摘We investigate the quantum coherence and quantum entanglement dynamics of a classical driven single atom coupled to a single-mode cavity. It is shown that the transformation between the atomic coherence and the atom-field entanglement exists, and can be improved by adjusting the classical driving field. The joint evolution of two identical single-body systems is also studied. The results show the quantum coherence transfers among composite subsystems, and the coherence conservation of composite subsystems is obtained. Moreover, the classical driving field can be used to suppress the decay of the coherence and entanglement, owing to considering the leaky cavity. The non-Markovian dynamics of the system is also discussed finally.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11574117 and11604131)
文摘By using a two-dimensional Monte-Carlo classical ensemble method, we investigate the double ionization（DI） process of the CS_2 molecule with different bond lengths in an 800-nm intense laser field. The double ionization probability presents a ＂knee＂ structure with equilibrium internuclear distance R = 2.9245 a.u.（a.u. is short for atomic unit）. As the bond length of CS increases, the DI probability is enhanced and the ＂knee＂ structure becomes less obvious. In addition,the momentum distribution of double ionized electrons is also investigated, which shows the momentum mostly distributed in the first and third quadrants with equilibrium internuclear distance R = 2.9245 a.u. As the bond length of CS increases,the electron momentum becomes evenly distributed in the four quadrants. Furthermore, the energy distributions and the corresponding trajectories of the double-ionized electrons versus time are also demonstrated, which show that the bond length of CS in the CS_2 molecule plays a key role in the DI process.
文摘A classical field theory for a Schrodinger equation with a non-Hermitian Hamiltonian describing a particle with position-dependent mass has been recently advanced by Nobre and Rego-Monteiro(NR)[Phys.Rev.A 88(2013)032105].This field theory is based on a variational principle involving the wavefunction Ψ(x,t) and an auxiliary fieldΦ{x,t).It is here shown that the relation between the dynamics of the auxiliary field Φ(x,t) and that of the original wavefunction Ψ(x,t) is deeper than suggested by the NR approach.Indeed,we formulate a variational principle for the aforementioned Schrodinger equation which is based solely on the wavefunction Ψ(x,t).A continuity equation for an appropriately defined probability density,and the concomitant preservation of the norm,follows from this variational principle via Noether's theorem.Moreover,the norm-conservation law obtained by NR is reinterpreted as tie preservation of the inner product between pairs of solutions of the variable mass Schrodinger equation.
文摘We propose two physical schemes, which can teleport unknown atomic entangled states from user A （Alice） to user B （Bob） via GHZ class states as quantum channel The two schemes are both based on cavity QED techniques. In the two schemes, teleportation and distillation procedures can be realized simultaneously. The second teleportation scheme is more advantageous than the first one.
基金supported by the National Natural Science Foundation of China (Nos. 11065007 and 11047115)the Scientific Research Foundation of Jiangxi Provincial Department of Education (Nos. GJJ10135 and GJJ09504)the Foundation of Talent of Jinggang of Jiangxi Province (No. 2008DQ00400)
文摘We investigate the entanglement dynamics of a quantum system consisting of two two-level atoms in a cavity with classical driving fields in the presence of white noise. The cavity is initially prepared in the vacuum state. Generally, the entanglement of two atoms decreases with the intensity of the thermal fields and the coupling strength of the two-level atoms to the thermal fields. However, we find that the entanglement of the quantum system can be enhanced by adjusting the frequency and the strength of the classical driving fields in the presence of white noise.
文摘We generalize several well known quantum equations to a Tsallis’ q-scenario, and provide a quantum version of some classical fields associated with them in the recent literature. We refer to the q-Schro¨dinger, q-KleinGordon, q-Dirac, and q-Proca equations advanced in, respectively, Phys. Rev. Lett. 106, 140601(2011), EPL 118,61004(2017) and references therein. We also introduce here equations corresponding to q-Yang-Mills fields, both in the Abelian and non-Abelian instances. We show how to define the q-quantum field theories corresponding to the above equations, introduce the pertinent actions, and obtain equations of motion via the minimum action principle.These q-fields are meaningful at very high energies(Te V scale) for q = 1.15, high energies(Ge V scale) for q = 1.001,and low energies(Me V scale) for q =1.000001 [Nucl. Phys. A 955(2016) 16 and references therein].(See the ALICE experiment at the LHC). Surprisingly enough, these q-fields are simultaneously q-exponential functions of the usual linear fields’ logarithms.
文摘Let G be a classical group over an arbitrary field F,acting on an n-dimensional vector space V=V(n,F)over a field F.In this paper,we classify the maximal subgroups of G,which normalizes a solvable subgroup N of GL(L,F)not lying in F^(*)1_(V).
基金funded by Scientific Research Project of Beijing Educational Committee(No.KM202110028004).
文摘Let G be a classical group over an arbitrary field F,acting on an n-dimensional F-space V=V(n,F).All those maximal subgroups of G are classified each of which normalizes a solvable subgroup N of GL(V/F)not lying in F^(*)1v.
基金supported by the National Natural Science Foundation of China (Grant No. 11735001)supported by the National Youth Fund (Grant No. 12105289)+1 种基金the UCAS Program of Special Research Associatethe Internal Funds of the KITS。
文摘We study the approaches to two-dimensional integrable field theories via a six-dimensional(6 D) holomorphic Chern-Simons theory defined on twistor space. Under symmetry reduction, it reduces to a 4 D Chern-Simons theory, while under solving along fibres it leads to a four-dimensional(4 D) integrable theory, the anti-self-dual Yang-Mills or its generalizations. From both 4 D theories, various two-dimensional integrable field theories can be obtained. In this work, we try to investigate several twodimensional integrable deformations in this framework. We find that the λ-deformation, the rational η-deformation, and the generalized λ-deformation can not be realized from the 4 D integrable model approach, even though they could be obtained from the 4 D Chern-Simons theory. The obstacle stems from the incompatibility between the symmetry reduction and the boundary conditions. Nevertheless, we show that a coupled theory of the λ-deformation and the η-deformation in the trigonometric description could be obtained from the 6 D theory in both ways, by considering the case that(3, 0)-form in the 6 D theory is allowed to have zeros.
