A Realization of Yangian and Its Applications to the Bi-spin System in an External Magnetic Field

Transcription

1 Commun. Theor. Phys. Beijing, China) ) pp. 1 5 c International Academic Publishers Vol. 39, No. 1, January 15, 003 A Realization of Yangian and Its Applications to the Bi-spin System in an External Magnetic Field JIN Shuo, 1,, XUE Kang, 3 and XIE Bing-Hao 1, 1 Theoretical Physics Division, Nankai Institute of Mathematics, Nankai University, Tianjin , China Liuhui Center for Applied Mathematics, Nankai University and Tianjin University, Tianjin , China 3 Physics Department, Northeast Normal University, Changchun 1300, China Received May 17, 00; Revised July 6, 00) Abstract Yangian Y sl)) is realized in the bi-spin system coupled with a time-dependent external magnetic field. It is shown that Y sl)) generators can describe the transitions between the spin triplet and the spin singlet that evolve with time. Furthermore, new transition operators between the states with Berry phase factor and those between the states of nuclear magnetic resonance are presented. PACS numbers: 0.0.-a, w Key words: Yangian, bi-spin system, transition operator 1 Introduction Yangian algebras were established by Drinfeld [1,] based on the investigation of Yang Baxter equation. In recent years, many works in studying Yangian and its applications have been made, including the Yangian symmetry in quantum integrable models such as Haldane Shastry model, [3] Calogero Sutherland model, [] and Hubbard model [5,6] ) and the realization of Yangian in quantum mechanics. [7 9] In quantum mechanics, the Yangian associated with sl) called Y sl)) has been realized in angular momentum quantum mechanics systems, [7] hydrogen atom [8,9] and other systems. The sense of Yangian generators transition operator has also been known. This is natural because Yangian algebras belong to Hopf algebras and have Lie algebras as their subalgebras. It was pointed out in Ref. [7] that, Y sl)) can be constructed in a bi-spin system with spin Ŝ1 and Ŝ Ŝk is the k-th spin operator). It has been known that the Yangian generators {Î, Ĵ can realize the transitions between the spin triplet and the spin singlet, which are the bases of quantum states of the bi-spin system each S k has spin-1/) that does not evolve with time. An interesting problem arises: if this system evolves with time, e.g. it is coupled with a time-dependent external magnetic field, what will happen for the transition operators. A physical picture of the transition between the states is useful to help us understand this time-dependent problem. In this paper, we will consider this issue. First, we will illustrate that Y sl)) generators {Ît), J ˆ t) still play the role of transition operators. Then, we discuss the transitions between the states with Berry phase factor of this system under the adiabatic condition. And finally, using the combination of the Y sl)) generators {Ît), J ˆ t), the transition operators between the nuclear magnetic resonance NMR) states are also obtained. Realization of Y sl)) and Transition Operators for the Bi-Spin System in an External Magnetic Field Y sl)) is formed by a set of operators {Î,Ĵ obeying the commutation relations: [1,7] [Îα, Îβ] = iɛ αβγ Î γ, [Îα, Ĵβ] = iɛ αβγ Ĵ γ α, β, γ = 1,, 3), [Ĵ±, [Ĵ3, Ĵ±]] = αĴ±Î3 αĴ3), [Ĵ3, [Ĵ+, Ĵ ]] = Î3Î+Ĵ Ĵ+Î ). 1) Î stands for the generators of sl). Hereafter for any operators ±,  ± = Â1±i  = Î, Ĵ) is understood. In a bi-spin system with spin Ŝ1 and Ŝ, the Y sl)) generators take the form: [7] Î Ŝ = Ŝ1 + Ŝ, Ĵ = µŝ1 Ŝ ) + ih Ŝ1 Ŝ, ) where µ and h are arbitrary parameters. In this paper, we consider the case that the two spins are all equal to 1/. Then direct calculation shows that the Yangian generators Ĵα α = ±, 3) describe the transitions between the spin triplet S = 1) X 11 =, X 10 = 1 + ), X 1 1 =, 3) The project supported by National Natural Science Foundation of China jinshuo@eyou.com

2 JIN Shuo, XUE Kang, and XIE Bing-Hao Vol. 39 and the spin singlet S = 0) X 00 = 1 ). ) The transition relations are Ĵ + X 1 1 = µ h Ĵ 3 X 10 = µ h ) ) Ĵ X 11 = µ h ) Ĵ + X 00 = X 11, Ĵ 3 X 00 = µ + h ) X 10, Ĵ X 00 = X ) Obviously the Lie algebra generators α can only describe the transitions between the spin triplet, but Ĵα α = ±, 3) can realize the transitions between the states with different Lie-algebra weights. The Hamiltonian of the bi-spin system coupled with a time-dependent external field Bt) reads Ĥt) = 1 Ŝ1 Ŝ γbt) Ŝ, 6) where γ is gyromagnetic ratio. To study this system, we firstly write a simple Hamiltonian describing the bi-spin system in a steady magnetic field as follows: Ĥ = 1 Ŝ1 Ŝ gŝ3, 7) where g is a constant. It is easy to see that the eigenstates of Ĥ are still the spin singlet and the spin triplet. So Yangian generators {Î, Ĵ can work well in this system. An interesting picture will appear if the magnitude of the magnetic field Bt) is chosen to be a time-independent constant B 0. Comparing Eq. 7) with Eq. 6), we find that through introducing the following unitary transformation operator Ût) = Û i t), i=1 Û i ) 1/ t) = Gt) Ŝ i i = 1, ), 8) B 0 + B 3 t)b 0 where Gt) = B 1 t), B t), B 3 t) + B 0 ), 9) there exists the following transformation relation Ĥt) = Ût)ĤÛ 1 t), 10) when we choose g = γb 0. From Eq. 10), it is shown that the eigenvalues of Ĥt) are the same as those of Ĥ, and the eigenstate X jmt) of Ĥt) can be got from the transformation of the eigenstate X jm of Ĥ, i.e., X jm t) = Ût) X jm jm = 11, 10, 1 1, 00), 11) where X jm is the spin triplet Eqs. 3)) or the spin singlet Eq. )). Through the unitary transformation Ût), the spin realization of Y sl)) has the time-dependent generators Ît) = Ût)ÎÛ 1 t), ˆ J t) = Ût)ĴÛ 1 t), 1) which can be verified still to satisfy the definition Eqs. 1) of Y sl)). Jα ˆ t) α = ±, 3) are the transition operators between the spin triplet S = 1) { X 11 t), X 10 t), X 1 1 t) and the spin singlet S = 0) X 00 t) in a time-dependent magnetic field, and the transition relations take the same form as Eqs. 5) J ˆ + t) X 1 1 t) = µ h ) X 00 t), 3 t) X 10 t) = µ h ) X 00 t), t) X 11 t) = µ h ) X 00 t), J ˆ + t) X 00 t) = X 11 t), 3 t) X 00 t) = µ + h ) X 10 t), t) X 00 t) = X 1 1 t). 13) In fact, the generators Ît) and J ˆ t) vary with Ût) because the selection of Ût) is not exclusive. The choice of Ût) does not affect the action of generators of Yangian Y sl)), so we can choose the unitary operator Ût) as simple as possible. Taking advantages of the above results, we will give two physical applications in the following discussions. 3 Transition Operators Between the States with Berry Phase Factor It has been shown that Berry phase [10] plays a fundamental and important role in quantum mechanics in the past two decades. Berry phase can be verified by experiments, [11] and it cannot be neglected in many physics lessons. Recently, Berry phase of the bi-spin system coupled with an external magnetic field has been studied. [1,13] Now we will give the transition operators between the states with Berry phase factor. Consider the bi-spin system in a rotating magnetic field described by B 1 t) = B 0 sin θ cos ω 0 t, B t) = B 0 sin θ sin ω 0 t, B 3 t) = B 0 cos θ, 1)

3 No. 1 A Realization of Yangian and Its Applications to the Bi-spin System in an External Magnetic Field 3 where the constant B 0 is the strength of the magnetic field. Substituting Eqs. 1) into Eqs. 8) and Eq. 11), we can give the eigenstates of Hamiltonian Eq. 6) that is the triplet S = 1) X 11 t) = cos θ) X e iω0t sin θ X cos θ) e iω0t X 1 1, X 10 t) = 1 sin θ e iω0t X 11 cos θ X 10 1 sin θ e iω0t X 1 1, X 1 1 t) = 1 1 cos θ) eiω0t X 11 1 e iω0t sin θ X cos θ) X 1 1, 15) and the singlet S = 0) X 00 t) = X ) Under the adiabatic condition, the state with Berry phase factor has the form { ψ jm t) = h E jmt exp {iγ jm t) X jm t), 17) where E jm is the eigenvalue of Hamiltonian Eq. 6) and is given by E 11 = 1 8 γb 0, E 10 = 1 8, E 1 1 = γb 0, E 00 = ) On the other hand, the Berry phase γ jm t) reads γ 11 t) = ω 0 1 cos θ)t, γ 1 1 t) = ω 0 1 cos θ)t, γ 10 t) = γ 00 t) = 0. 19) By comparing Eqs. 13) with Eq. 17), we immediately find that J ˆ + t) ψ 1 1 t) = µ h ) { exp ī h E 1 1 E 00 )t 3 t) ψ 10 t) = µ h ) { h E 10 E 00 )t ψ 00 t), exp {iγ 1 1 t) ψ 00 t), t) ψ 11 t) = µ h ) { h E 11 E 00 )t exp {iγ 11 t) ψ 00 t), J ˆ + t) ψ 00 t) = µ h ) { h E 11 E 00 )t exp { iγ 1 1 t) ψ 11 t), 3 t) ψ 00 t) = µ + h ) { ī exp h E 10 E 00 )t ψ 10 t), t) ψ 00 t) = { ī exp h E 1 1 E 00 )t exp { iγ 1 1 t) ψ 1 1 t). 0) We have solved the transition problems between the states with Berry phase factor by applying ˆ J α t) α = ±, 3). Transition Operators Between the States of NMR NMR is a very important experiment technique based on quantum mechanics. [1,15] It has made rapid progress since 195. Very recently, NMR is used to realize the geometric quantum computation. [16,17] The motivation for this section is to find the transition operators between the NMR states. We choose the same magnetic field and the Hamiltonian as that in the former sections. By solving the Schrödinger equation and utilizing the magnetic resonance condition MRC) ω 0 = γb 3, 1) we can get the states of NMR by choosing different initial states. These states are the time-dependent combination of the eigenstates of the Hamiltonian Eq. 6). The triplet S = 1) has the forms φ 11 t) = a 1 t) X 11 t) + a t) X 10 t) + a 3 t) X 1 1 t), φ 10 t) = b 1 t) X 11 t) + b t) X 10 t) + b 3 t) X 1 1 t), φ 1 1 t) = c 1 t) X 11 t) + c t) X 10 t) + c 3 t) X 1 1 t). )

4 JIN Shuo, XUE Kang, and XIE Bing-Hao Vol. 39 In the process of calculating, the eigenvalues of deformed wave functions under MRC Eq. 1) have the exact values E 11 = 1 8 γb 1, E 10 = 1 8, E 1 1 = γb 1, E 00 = ) The time-dependent coefficients in Eqs. ) are a 1 t) = 1 { exp ī h E 10t exp{iω 0 t cos θ cos ω 0 t + cos ω 0 t cos ω 1 t + i sin ω 0 t + i cos θ sin ω 0 t cos ω 1 t + i sin θ sin ω 1 t), a t) = 1 { h E 10t sin θ cos ω 0 t + i sin θ sin ω 0 t cos ω 1 t i cos θ sin ω 1 t), a 3 t) = 1 { exp ī h E 10t { iω 0 t cos θ cos ω 0 t cos ω 0 t cos ω 1 t i sin ω 0 t + i cos θ sin ω 0 t cos ω 1 t + i sin θ sin ω 1 t), b 1 t) = 1 { h E 10t exp{iω 0 ti cos ω 0 t sin ω 1 t cos θ sin ω 0 t sin ω 1 t + sin θ cos ω 1 t), { b t) = h E 10t sin θ sin ω 0 t sin ω 1 t + cos θ cos ω 1 t), b 3 t) = 1 { h E 10t exp{ iω 0 ti cos ω 0 t sin ω 1 t + cos θ sin ω 0 t sin ω 1 t sin θ cos ω 1 t), c 1 t) = 1 { exp ī h E 10t exp{iω 0 t cos θ cos ω 0 t + cos ω 0 t cos ω 1 t i sin ω 0 t + i cos θ sin ω 0 t cos ω 1 t + i sin θ sin ω 1 t), c t) = 1 { h E 10t sin θ cos ω 0 t i sin θ sin ω 0 t cos ω 1 t + i cos θ sin ω 1 t), c 3 t) = 1 { exp ī h E 10t exp{ iω 0 t cos θ cos ω 0 t cos ω 0 t cos ω 1 t + i sin ω 0 t + i cos θ sin ω 0 t cos ω 1 t + i sin θ sin ω 1 t), ) where ω 1 = γb 1. The singlet S = 0) is φ 00 t) = exp{ ie 00t X 00 t). 5) From Eqs. 13), ), and 5), after a rather long calculation we get the following relations t) φ 11 t) = a 1 t) µ h ) 00t φ 00 t), {a 3 t) J ˆ t) a 1 t) J ˆ + t) + a t) J ˆ 3 t) φ 00 t) = 00t φ 11 t), 3 t) φ 10 t) = b t) µ h ) 00t φ 00 t), {b 3 t) J ˆ t) b 1 t) J ˆ + t) + b t) J ˆ 3 t) φ 00 t) = 00t φ 10 t), J ˆ + t) φ 1 1 t) = c 3 t) µ h ) 00t φ 00 t), {c 3 t) J ˆ t) c 1 t) J ˆ + t) + c t) J ˆ 3 t) φ 00 t) = 00t φ 1 1 t). 6) From Eqs. 6), we can draw the conclusion that Jα ˆ t) α = ±, 3) and its combination construct the transition operators between the NMR states for simplicity, among the above relations, we have chosen the constants which can be modified to zero or 1 if it is possible). 5 Conclusions In summary, we get a time-dependent realization of Y sl)) in the bi-spin system coupled with a time-dependent

5 No. 1 A Realization of Yangian and Its Applications to the Bi-spin System in an External Magnetic Field 5 external magnetic field. Although we can verify that Y sl)) does not describe the symmetry of the system which we study because J ˆ t) does not commute with Ĥt), we concentrate on the transition function of Yangian. We find that the generators {Ît), J ˆ t) of Y sl)) can describe a new picture of transition between two quantum states at any time. For briefness, we have neglected the part of transitions that are described by the Lie algebra operators. As far as we know, many interesting investigations have relations with the bi-spin model coupled with a time-dependent external magnetic field, such as geometric quantum computation, [16,17] entanglement. [18] At last, we emphasize that Yangian algebras belong to Hopf algebras and take Lie algebras as their subalgebras, so the Yangian operators can connect the physical states with different Lie-algebra weights. It is reasonable to believe that more interesting Yangian realization and more useful physical applications should be found. Acknowledgments We thank Drs. Hong-Biao ZHANG and Hui JING for valuable discussions. References [1] V.G. Drinfeld, Soviet Math. Dokl ) 5. [] V.G. Drinfeld, Soviet Math. Dokl ) 1. [3] F.D.M. Haldane, Z.N.C. Ha, J.C. Talstra, D. Bernard, and V. Pasquier, Phys. Rev. Lett ) 01. [] M. Wadati, Phys. Rev. Lett ) 635. [5] D.B. Uglov and V.E. Korepin, Phys. Lett. A90 199) 38. [6] F. Göhmann and V. Inozemtsev, Phys. Lett. A1 1996) 161. [7] M.L. Ge, K. Xue, and Y.M. Cho, Phys. Lett. A9 1998) 358. [8] M.L. Ge, K. Xue, and Y.M. Cho, Phys. Lett. A ) 8. [9] C.M. Bai, M.L. Ge, and K. Xue, J. Stat. Phys ) 55. [10] M.V. Berry, Proc. R. Soc. Lond. A39 198) 5. [11] R.Y. Chiao and Y.S. Wu, Phys. Rev. Lett ) 933. [1] L.G. Yang and F.L. Yan, Phys. Lett. A65 000) 36. [13] F.L. Yan and L.G. Yang, Commun. Theor. Phys. Beijing, China) ) 57. [1] C.P. Slichter, Principles of Nuclear Magnetic Resonance, Springer-Verlag, 1978) nd ed. [15] C.T. Claude, K. Bernard, and L. Franch, Quantum Mechanics, France, Hermam, John Wiley and Sons. Inc. 1977). [16] J.A. Jones, V. Vedral, A. Ekert, and G. Castagnoli, Nature ) 869. [17] J.A. Jones, Progress in Nuclear Magnetic Resonance Spectroscopy ) 35. [18] R.G. Unanyan, N.V. Vitanov, and K. Bergmann, Phys. Rev. Lett )